/sci/ - Science & Math

>>12408192
all these retards who say 1/3 are literal trannies
it's 1/4 don't (You) me

100%

>>12408192
4/9
You pick a ball from *a* box

>>12408471
Wrong. See https://en.wikipedia.org/wiki/Boy_or_Girl_paradox
If your imaginary friend who wrote the test question is an illiterate retard who doesn't know English, they might SAY "at least """one""" is heads" but really they MEAN "at least """this particular coin""" is heads." And since it's not polite to make fun of retards, we know that the question is therefore ambiguous because the narrator of the question could literally be a retard, or maybe the typesetter was drunk and made a typo. We just don't know.
t. schizo

>>12408498
ahahahahaha

>>12408195
Explain to me why, I would have said 1/3...
Unless this is actually a riddle or you are stupid

>>12408192
this problem is unironically what is behind the measurement problem in QM.

>>12408498
Loser

>>12408195
this is completely wrong

>>12408498
It was never stated if a third coin was flipped behind your back.
Then even if you get tails in both hands, you could be told that at least one is heads!!!!!!!!!

Guys, the question is CLEARLY ambiguous. Think outside the boxerino!!!!!!

>>12408628
The "at least one is heads" is ambiguous. Cry more retard.

>>12408498
>they might SAY "at least """one""" is heads" but really they MEAN "at least """this particular coin""" is heads."

I'm pretty sure the problem reads as "at least one is HEADS". Literally.
>illiterate retard

>>12408644
Give me a single textbook probability problem where "at least" is used to refer to one single specific object.

>>12408192
1/2 because the only possible outcomes are 1 heads 1 tail or 2 heads

>>12408192
Probability space of flipping 2 coins
H H
H T
T H
T T
3 have at least one head, only 1 has both heads.
Therefore 1/3

>>12408759
two possible outcomes don't necessarily imply 50/50 probability

>>12408821
One is already known as heads, So the left column is irrelevant. You're only flipping one coin.

50% probability

>>12408192
94.31 %

>>12408192
101%

>>12408925
There's also the case where the first one lands on tails so the probability of the second one landing on heads is 100%
So
H 50% H 50%
H 50% T 50%
T 50% H 100%

>>12408652
See >>12408498

>>12408652
>give me a textbook problem worded amiguously

>>12409046
>worded amiguously
I never said that.
I said a problem where "at least" is used to refer to one single specific object.
This would mean "at least" doesn't have one universal meaning accross textbook probability problems.
This would prove OP's problem is worded ambiguously.
I am asking for examples of this.
And not an example where you hallucinate imaginary friends and where the predicate is somehow biased by superfluous context.
>illiterate retard

>>12408192
156/731.
Dont ask me how i know it, okay?

>>12409025
There's the answer.
Also who the fuck cares about OP's problem? What a waste of time.

>>12409077
>I never said that.
And I never said "at least one is heads" refers to a specific coin, I said it's ambiguous. A textbook would be written in non-ambiguous language.

>This would mean "at least" doesn't have one universal meaning accross textbook probability problems.
And?

>This would prove OP's problem is worded ambiguously.
No it wouldn't. The proof that OP's problem is ambiguous is that it can be interpreted in two different ways within the context of the problem. This proof has already been given to you and there are several papers discussing it. Yet you continue to deny it for no reason other than deliberate stupidity.

>I am asking for examples of this.
What is "this?"

>>12408652
Mr. Smith has new neighbors with two children. Mr. Smith tells you he met one of the children so he knows at least one of them is a boy. What is the chance both are boys?

>>12409448
>And I never said "at least one is heads" refers to a specific coin
I know that. But if the phrase, as used in textbooks, always has the same meaning, you're expected to interpret it the same way when you come accross it.
I am asking for examples of problems where the phrase has the "second" meaning you mentioned, which then would make "at least" ambiguous when written without specification.

>The proof that OP's problem is ambiguous is that it can be interpreted in two different ways within the context of the problem.
And I'm saying there is only one way it can be interpreted, unless you show examples of the other way.

And not an example where you hallucinate imaginary friends and where the predicate is somehow biased by superfluous context.

>>12408652
"At least" simply refers to a lack of knowledge about the entire set. It has nothing to do with multiple objects or specific objects you have knowledge of.

>>12409494
>But if the phrase, as used in textbooks, always has the same meaning, you're expected to interpret it the same way when you come accross it.
Same way as what? You're not making much sense. I'm not aware of any other type of problem in which this ambiguity occurs, yet you act as if every textbook has you interpret the phrase one way and not the other way. If this were actually true then it would just mean textbooks are full of ambiguous problems. But I doubt they are. This has nothing to do with the problem at hand.

>I am asking for examples of problems where the phrase has the "second" meaning you mentioned, which then would make "at least" ambiguous when written without specification.
See >>12409495

>And I'm saying there is only one way it can be interpreted
Which is proven wrong by several papers cited in the wiki article. You have no argument.

>>12409494
>But if the phrase, as used in textbooks, always has the same meaning, you're expected to interpret it the same way when you come accross it.
Same way as what? You're not making much sense. I'm not aware of any other type of problem in which this ambiguity occurs, yet you act as if every textbook has you interpret the phrase one way and not the other way. If this were actually true then it would just mean textbooks are full of ambiguous problems. But I doubt they are. This has nothing to do with the problem at hand.

>I am asking for examples of problems where the phrase has the "second" meaning you mentioned, which then would make "at least" ambiguous when written without specification.
See >>12409482

>And I'm saying there is only one way it can be interpreted
Which is proven wrong by several papers cited in the wiki article. You have no argument.

>>12409494
Showing problems in which a phrase can only be interpreted one way (which you haven't done by the way) does not show a different problem must be interpreted one way.

>>12409495
Let X be the number of things you are counting. in this case it's coins.

At least one coin X ≥ 1

We don't what is the value of X.
But we know it's not smaller than one.

>>12409539
>if the phrase, as used in textbooks, always has the same meaning, you're expected to interpret it the same way when you come accross it.
>textbooks and professors assume you're not an autistic baby who needs every little detail pedantically explained to them because they can't interpret a predicate the same way you would with every other math problem.

>>12409548
Yeah, so "at least" refers to a lack of knowledge about the rest of the set of coins. We know somehow that there is one coin, but we don't know it's the only coin. So we say "at least one" instead of "one."

>>12409558
So you're saying if you came across this problem: >>12409482 you would say the answer is 1/3 because you're "not an autistic baby?" Would you blame your professors and textbooks when you get marked wrong?

LOL. Ignoring context and ambiguity doesn't make you smart.

>>12409482
Different problem, because some information is given by an entity that operates inside the context of the problem.
This fact renders the information possibly biased which means it needs to be contextualized so one can assess condtional porbability of events.
Note that the person was mentionned and not hallucinated.
Different problem.

>>12409528
>Same way as what?
Since we are talking about the phrase "at least one", See >>12409548

>Which is proven wrong by several papers
Those papers make the distinction between different interpretations based on the context.
"The paradox occurs when it is not known how the statement "at least one is a boy" was generated. "
The 1/2 answer applies to cases when the information given is limited by context, like in your Mr. Smith example.
This doesn't apply here.

In OP's problem, information is provided by the predicate, and is assumed as a hypothesis.
You are not told there is someone else that observes the results, who may give you biased info.
The information is given to you per se and is assumed as a fact.

>>12409565
>Yeah, so "at least" refers to a lack of knowledge about the rest of the set of coins. We know somehow that there is one coin, but we don't know it's the only coin. So we say "at least one" instead of "one."
Yes.

In OP's case, a flip that satisfies "at least one" has a certain probability.
P(X ≥ 1) = 3/4.
This is the real result as shown empirically.
Of course if you were given info about "at least one" being heads by an enitity that operates inside the problem, you would have to consider how the information was obtained.
But there is no mention of such thing.

>if you came across this problem [...] you would say the answer is 1/3
I wouldn't, the answer would then be 1/2. But again, that's a different problem.

>Would you blame your professors and textbooks when you get marked wrong?
I'd certainly get marked wrong if I tried to outsmart them by doing word puzzles when they want me to do math

>Ignoring context
What context is there besides "You flip two coins"? You're hallucinating.

>>12409605
>Different problem
Yeah, that's what you asked for.

>>12408498
>Many people[who?] argued strongly for both sides with a great deal of confidence, sometimes showing disdain for those who took the opposing view[citation needed].
Which one of you wrote this?

>>12409605
>Since we are talking about the phrase "at least one", See >>12409548
That doesn't actually clarify anything. How does it prove your interpretation is correct?

>"The paradox occurs when it is not known how the statement "at least one is a boy" was generated. "
So you know how the statement was generated?

>The 1/2 answer applies to cases when the information given is limited by context, like in your Mr. Smith example.
So does the 1/3 answer. You're not actually making an argument.

>In OP's problem, information is provided by the predicate, and is assumed as a hypothesis.
What information?

>You are not told there is someone else that observes the results, who may give you biased info.
Yes, that's what makes it a "paradox." Why are you pretending the source you're citing agrees with you when it clearly doesn't?

>The information is given to you per se and is assumed as a fact.
Yeah, the problem is the information is ambiguous. It's given and assumed as fact in either interpretation. Are you really still not grasping this or are you pretending to be retarded?

>>12408192
1/4

>>12408925
You don't know which of the coins will be heads though, so HT and TH have to be considered as separate events

>>12409899
Couple questions
1. Are you autistic? On the spectrum?
2. Are you schizophrenic?
3. NEET?
4. Completed a probability course? Or seriously studied the subject on your own?

>>12408925
But knowing one is heads when the other is tails happens twice as often as when the other is also heads.

>>12409899
>Yeah, that's what you asked for.
An example where the info is not given by a person in the context of the problem.
Because yes, that changes how you should interpret what they are saying, which has nothing to do with the problem here.

>How does it prove your interpretation is correct?
Because it's the one everybody on earth uses with the type of information given (no imaginary friends).
If you you ever did any kind of probability course this would be evident.

>So you know how the statement was generated?
It is written.
>illiterate retard

>applies to cases when the information given is limited by context
>So does the 1/3 answer
How? Where?
Again, what context is there besides "You flip two coins"?

>What information?
Idk, maybe read the problem

>Yes, that's what makes it a "paradox."
No paradox if there is no contextual loophole to exploit

>the problem is the information is ambiguous
only ambiguous if there's mention of context that could change the meaning of the info.
If a problem becomes ambiguous simply because it doesn't explicitely mention the context of the problem, you could exploit this and come up with any kind of answer, see >>12408628

I'm getting tired of this. I'm going to stop paying attention to this thread, this is becoming ridiculous.

>>12409650
>In OP's case, a flip that satisfies "at least one" has a certain probability.
>P(X ≥ 1) = 3/4.
It depends what you mean by "at least one is heads."

>Of course if you were given info about "at least one" being heads by an enitity that operates inside the problem, you would have to consider how the information was obtained.
You have to consider how the information was obtained either way, because there is more than one way to reach the conclusion "at least one is heads" and it affects the answer. If you don't want to think of it as how the information was obtained, then think of it as whether "at least one is heads" refers to a specific coin or not.

>But there is no mention of such thing.
Then it's ambiguous.

>I wouldn't, the answer would then be 1/2.
How is that possible? "At least one" can only have one interpretation. It can ONLY have probability 3/4.

So P(BB|A) = (1/4) P(A|BB) / (3/4) = P(A|BB)/3

Thus it's IMPOSSIBLE for the answer to be 1/2. The answer must be between 0 and 1/3.

>What context is there besides "You flip two coins"?
The context is Mr. Smith only saw one boy.

>>12410019
Do you have an argument? No? OK then read this if you're too stubborn to accept that you're wrong: https://en.wikipedia.org/wiki/Boy_or_Girl_paradox

>>12410099
this tbqfh

>>12410060
>An example where the info is not given by a person in the context of the problem.
No, that's not what you asked for, and if you did it would be an irrelevant request. If that info was not given it would be an ambiguous statement.

>Give me a single textbook probability problem where "at least" is used to refer to one single specific object.

>>12408192

>>12410060
>Because it's the one everybody on earth uses with the type of information given (no imaginary friends).
That's empirically wrong. See https://en.wikipedia.org/wiki/Boy_or_Girl_paradox

>If you you ever did any kind of probability course this would be evident.
I have. And probability theorists disagree with you. Get over it.

>It is written.
That's not how it's generated. Go read what you quoted again.

>How? Where?
In the context where "at least one heads" is a non-specific statement.

>Again, what context is there besides "You flip two coins"?
You're incorrectly assuming that the (lack of) context gives 1/3 as the answer.

>Idk
I know.

>No paradox if there is no contextual loophole to exploit
The "paradox" comes from lack of context, not context.

>only ambiguous if there's mention of context that could change the meaning of the info.
Only ambiguous because there is no context.

>If a problem becomes ambiguous simply because it doesn't explicitely mention the context of the problem, you could exploit this and come up with any kind of answer
You can, in cases where context crucial to answering the question is missing. That's what you're exactly what you're doing by demanding that 1/3 is the answer.

>see >>12408628
I see a complete non sequitur, since the problem is ambiguous already.

>>12410306
>probability theorists disagree with you

>Gardner initially gave the answers 1/2 and 1/3, respectively, but later acknowledged that the second question was ambiguous
>Martin Gardner (October 21, 1914 – May 22, 2010) was an American popular mathematics and popular science writer, with interests also encompassing scientific skepticism, micromagic, philosophy, religion, and literature—especially the writings of Lewis Carroll, L. Frank Baum, and G. K. Chesterton.

>The ambiguity, depending on the exact wording and possible assumptions, was confirmed by Bar-Hillel and Falk,[4] and Nickerson.[5]
>Maya Bar-Hillel (Hebrew: מיה בר-הלל, born 1943)[1] is a professor emeritus of psychology at the Hebrew University of Jerusalem.[2]
>Roma Falk (Oren) ( 1932 - August 15, 2020 ) was a professor of psychology at the Hebrew University of Jerusalem
>Raymond S. Nickerson is an American psychologist and author.

just look at those deluded psychologists trying to do math. made my day.

>>12408192
so you flip both coins
and if neither had been heads you'd reflip?

its a subset of flips, not flipping one since the other outcome is fixed?
Is that the "ambiguity"?

>>12409482
HAHAHA fucking retard schizo.

>>12410306
I think this schizo is on to something, guys... Don't you see?

"You flip two coins, at least one is heads."

Yes, obviously any English speaker with an IQ in the Fahrenheit range of liquid water knows that the second clause of the sentence refers to the first clause of the sentence:
"You flip two coins, at least one [of the ref. "two coins"] is heads."

BUT... Wait for this..... If your imaginary friend who wrote the test is either an illiterate retard, a troll, or a hallucinating schizo, they might instead be INDIRECTLY referencing a different, imaginary clause in a different, imaginary sentence:
"You flip two coins, at least one [of the imag. ref. "one specific coin"] is heads."

Don’t you see? It’s EXACTLY like this wikipedia article I read in prison. There's no difference!

>>12410306
"You flip two coins, the first one is HEADS. What is the probability that both are HEADS?"

is this ambiguous?

>>12411004
also unambiguous

>>12408192
50%?

what if it lands on its edge you homos

ever thought of that?

>The probability of a coin landing on its edge are completely dependent on the type of coin being flipped. Some coins have rounded edges, which would make it impossible for the coin to land and stay upright when flipped onto a hard flat surface, unless the coin was physically modified.

On the other hand, thicker coins that are designed with a flat edge, like older British one pound coins or American nickels (both pictured below) have a small, but non-zero probability of landing on the edge of the coin. In which case, the number of outcomes for any coin flip is three (heads, tails, edge) and the probability can no longer be 50/50 due to the third possible outcome.

>>12410786
https://math.dartmouth.edu/~prob/prob/prob.pdf
>Charles M. Grinstead, Professor, Department of Mathematics and Statistics, Swarthmore College
>J. Laurie Snell, Professor, Department of Mathematics, Dartmouth College

>>12411004
No. What is your point?

>>12411297
HAHAHAHAHAHA
Every text this retard links actually ends up shitting all over his own schizo hallucination that there's any ambiguity at all in OP.

cf. p. 151, x. 9(a)(b); pp.175,176, xx. 4.25,4.26

Lmfao what an absolute retard.

>>12411297
??? authors of the text clearly consider pic related to be an unambiguous problem worthy of being included in an introductory probability course

>>12411344
>>12411385
It says it's ambiguous, the answer can be 1/2 or 1/3 depending on your assumptions. Get over it.

>>12411390
No, you absolute fucking idiot. It says nothing of the sort.

The authors of this pdf literally write an exercise with what they consider to be two unambiguously different questions:
>What is the probability that a family of two children has
>(a) two boys given that it has at least one boy?
>(b) two boys given that the first child is a boy?

And in the section that discusses ambiguity, they write this:
>For example, they show that 1/2 is the correct answer for the following scenario.
>Mr. Smith is the father of two. We meet him walking along the street with a young boy whom he proudly introduces as his son. What is the probability that Mr. Smith’s other child is also a boy?
HAHAHAHAHA NO SHIT SHERLOCK

You didn't even read your own source material you fucking absolute fucking retard.
LMFAO

>>12408192
Conditioned that at least 1 is heads the probability of both being heads is 0.5.

>>12411407
>The authors of this pdf literally write an exercise with what they consider to be two unambiguously different questions:
Where does it say the questions are unambiguous?

And in the section that discusses ambiguity, they write this:
>For example, they show that 1/2 is the correct answer for the following scenario.
And how does 1/2 being the answer for another question show the first is unambiguous? You lost.

>>12411407
It says the question requires more detail to resolve the paradox. You can't read.

>>12411437
HAHAHAHA Squirm and flail you fucking worm brained retard.
Literally blown the fuck out by your own source material.

>>12411437
>Where does it say the questions are unambiguous?
>the authors never explicitly stated that they would include only unambiguous exercises
>"you lost"

>>12411448
Not an argument. You lose.

>>12411444
Lmfao schizo, now you're literally hallucinating shit again. Cite your hallucination in the text.

>>12411451
HAHAHAHAHAHAHA R

>>12411449
See >>12411451

>>12408195
Makes sense to me. OP pic never said that the first coin would be guaranteed to be heads.

So basically there are 4 scenarios

HH
HT
TH
TT

TT is out because one of the coin is heads. That leaves 3 possible scenarios, out of which one is HH

Hence the likelyhood is 1/3 for HH!

>>12410194
Hesus chitinus thats fucken ugly.

>>12411567
So basically there are 2 scenarios

HH
HT

Since we know a coin is heads. That leaves only one other coin which has a 50% chance of being heads.

Hence the likelyhood is 1/2 for HH!

>>12411840
So basically there are 2 scenarios

H_
_H

Since we know a coin is heads.
But the only way the other coin can be tails is if we know it's not the heads coin.
But we CAN'T know which coin is the heads coin, so the other coin has a 0% chance of being tails.

Hence the likelihood is 1 for HH!

>>12408192
Why is it permissible to assume that the coins have chronological order and are differentiable from each other at all times? It's not stated in the question.

>>12408192
50% obviously. is that a trick question?

21/37

>>12411840
Put it this way, you flip two coins A and B ”AB”. This gives us the following scenarios:

HH
HT
TH
TT

However from the text it is evident that at least one of the coins is heads. That means that we actually have three scenarios.

HH
HT
TH

Thus the likelyhood of HH is 1/3. I’m a genious!

100%

>>12412018
Jeszcze jak.

>>12411390
a) can only be 1/3
b) can only be 1/2

The information give at a) means three possible scenarios and the information at b) means two possible scenarios.

P(aUbUc) with a=h and h, b=h and t c=t and h?

>>12412338
However from the text it is evident that at least one of the coins is heads. That means that we actually have two scenarios.

HH
HT

Thus the likelyhood of HH is 1/2.

>>12412401
OK and?

>>12412472
However from the text we don't know which of the coins is heads. That means that we actually have two scenarios.

H_
_H

Thus the likelihood is 1.

>>12412472
You have 3 coin 1 and coin 2 can be HH, HT, TH

>>12412472
It doesn’t say which coin is heads though. ”TH” is also possible, not just HT.

HH
HT
TH

two possible outcomes doesn't imply 50/50 probability. it's possible to treat the tosses as unordered and declare HT = TH, that leaves only two options

TH
HH

but they have different likelihoods of happenings. clearly TH happens more often than HH.

>>12412496
>However from the text we don't know which of the coins is heads.
Whichever one it is, the other has a 50% chance of being heads.

>>12412530
Mmmhmm, but HT and TH are the same, you're just switching the position of the known coin.

>>12412535
>It doesn’t say which coin is heads though.
It's whichever one "at least one" refers to. Only the other coin matters.

>>12412591
>two possible outcomes doesn't imply 50/50 probability.
Right, but in this case the two possible outcomes are equally likely. Either the other coin is a heads or tails, with equal probability.

>>12412650
There are two known coins A and B. It is not known what coin will flip H or if both will. It is easier to understand the problem if you think what combinations are possible in a coin flip with two coins and then remove the TT since that is excluded in the problem. Then you are left with TT, TH and HT ie 1/3.

>>12412655
you're choosing between HH and TH but TH happens more often than HH. are you saying they're equally likely to happen?

>>12412643
And you have a 0% chance of knowing which coin is the “other” so that info is 100% irrelevant.

>>12412732
>It is not known what coin will flip H or if both will.
It already happened.

>It is easier to understand the problem if you think what combinations are possible in a coin flip with two coins and then remove the TT since that is excluded in the problem.
It doesn't matter how you order the coins. We know one coin is heads so the other coin has a 50% chance of also being heads. If you are not ordering by which coin is known and which coin isn't then you get

Hh
hH
Ht
tH

where the uppercase coin is the known coin and the lowercase coin is the unknown coin.

>>12412800
>you're choosing between HH and TH but TH happens more often than HH.
No, it happens at the same rate if the second coin is the known coin.

>>12412823
>And you have a 0% chance of knowing which coin is the “other”
It doesn't matter. Whichever one it is the probability is the same. That info is 100% irrelevant.

>>12412858
you don't know which coin is the known coin. there's no "the second coin"

>>12408192
50 50

its either or it is nt

>>12412858
You’re both wrong, it’s 1, but see this
>>12412823
You have a 0% chance of knowing which coin is heads

>>12408192
0.9999999999...
/
3

>>12412864
>you don't know which coin is the known coin.
Doesn't matter. See >>12412850

>there's no "the second coin"
You misunderstand. I'm taking about the second coin in your ordering: TH, HH. The known coin in your ordering has to be the second coin because you allow the first coin to be tails.

>>12412883
You have 0% chance of knowing which coin is the known coin. You’re trying to bring the future into the past, it’s not 1/2 it’s 1.

>>12412904
>You have 0% chance of knowing which coin is the known coin.
Wrong and irrelevant. What if you just guess? And as I already showed, you don't need to know which coin is the known coin. The other coin has a 50% chance of being heads regardless of which coin it is.

>>12412936
Wrong, you have a 0% chance of knowing the “other” coin. You can’t calculate anything about the “other” coin, because you can’t know what it is, you’re just making things up.

>>12412975
>know what it is
*know which it is

>>12412975
>Wrong, you have a 0% chance of knowing the “other” coin.
Wrong and irrelevant.

>You can’t calculate anything about the “other” coin
I just did. Deal with it.

>because you can’t know what it is
Irrelevant.

>>12412987
You did nothing, you pretended you knew which coin is “other” and came up a fake answer based on your fake knowledge

>>12412996
>You did nothing, you pretended you knew which coin is “other”
No I didn't. I don't see the point in arguing with you if you can't even read English.

>>12412936
>The other coin has a 50% chance of being heads regardless of which coin it is.
that's true, but that's not what's being asked. the fact that you don't know which coin is heads has to be included.

>>12413009
You came up with a fake description of the two scenarios.
it’s not
TH
HH
it’s
_H
H_

>>12413026
>that's true, but that's not what's being asked.
It is. If the other coin is heads then both are heads.

>the fact that you don't know which coin is heads has to be included.
Doesn't matter.

>>12413028
>You came up with a fake description of the two scenarios.
How is it fake?

>it’s
>_H
>H_
This is the same thing. The unknown coin has a 50% chance of being heads in either case:

Hh
hH
Ht
tH

You flip two coins, one is gay, the other is gay. You are gay.

>>12412646
the coins are independant you degenerate moron

>>12413193
>Doesn't matter
It does, because you have to consider every case where a coin satisfies the condition, wherever it is.
Permutations matter

>>12413607
Nothing I said implies the coins are not independent. Are you really this stupid?

>>12413616
>It does, because you have to consider every case where a coin satisfies the condition, wherever it is.
I did.

Hh
hH
Ht
tH

>>12413736
Heads/heads can only happen in one way. It makes no sense to me to say Hh or hH.

>>12413197
Your new thing is also fake because the coins are both already flipped. You don't get to "know" which coin is "unknown" and then flip the "unknown" coin again.

There are only two scenarios, not four.
_H
H_

>>12414183
>Heads/heads can only happen in one way.
Right, but heads/heads and one coin being known can happen in two ways. You're ignoring permutations.

>>12414709
You can't "know" a coin.

>>12414654
>Your new thing is also fake because the coins are both already flipped.
Nothing I said contradicts that. It's also not a "new thing," it's exactly the same as

HH
HT

It's just that you took offense at ordering according to the known coin and the unknown coin even though by symmetry the ordering doesn't matter.

>_H
>H_
This is exactly the same as what I said. The unknown coin has a 50% chance of being heads. So the answer is 50%.

>>12414713
"At least one is heads" means one coin is known to be heads.

>>12414718
No, you never get to "know" which coin is the "unknown" coin. You're just making that up so you can pin down a new 50% "coin toss". There is no new coin toss.

There are two coins that have already been flipped and all you can ever know is that at least one is heads for sure, you can't know anything about the "other" one because you can't know which one is "other" one is.
_H
H_
The only possible answer is 100%

>>12414720
But you can never know which coin it is.

>>12414725
>No, you never get to "know" which coin is the "unknown" coin.
I didn't say you did, not does it matter. Nice reading.

>you can't know anything about the "other" one because you can't know which one is "other" one is.
Doesn't follow. We already know coins have a 50% chance of landing on heads.

I'm not responding to this gibberish again.

>>12414755
>I didn't say you did
>>12414718
>The unknown coin has a 50% chance
Liar

Easy proof that you're the one making up gibberish, look at this gibberish you wrote.
>>12413736
>Hh
>hH
This is nonsense unless you pretend you know which coin is "known" and which coin is "unknown", which you don't and can't.

>>12414786
>>The unknown coin has a 50% chance
This doesn't say i know which coin is unknown.

>This is nonsense unless you pretend you know which coin is "known" and which coin is "unknown"
Either coin is unknown or known. Look at it again. It's the left column restricted to capital letters? Is the right column. This is the last reply you get since you can't even read English correctly or make basic logical conclusions.

>>12408192
33% exactly
Anyone saying 1/3rd is retarded

>>12414830
It's 1/2. One coin is heads so the other has a 50%chance of also being heads.

>>12414816
It's completely fake, the two coins were already flipped.
Look at this gibberish.
>Hh
>hH
You're trying to pretend that the same result where two coins that were already flipped are both heads, are actually two different results.

The two coins are already flipped. You don't flip one coin and pause and "know" that it's heads and then move on to flipping the second coin. Both coins are already flipped.

>>12414837
No you retard because there's 3 cases which contain head

>>12409270
So close, its actually 157/731

>>12414830
Learn your fractions, it 29% exactly

>>12414837
There's no "the other one". you don't know which coin is heads.

>>12414709
If you flip two coins HH can only happen in one way. If you know which coin is H then the probability of HH is 1/2. If you don’t know which coin is H the probability is 1/3. There is only one way for HH to happen, not two. This is common sense. If you were to flip two coins a thousand times and exclude all tosses that ends in TT then 1/3 would be HH. The information given to the problem is that the toss isn’t TT, no other information is given.

>>12414841
>You're trying to pretend that the same result where two coins that were already flipped are both heads, are actually two different results.
They are only two different results because you insist on treating TH and HT differently even though by symmetry they are the same. There is a 50% chance the known heads is on the right or on the left.

>The two coins are already flipped.
Nothing I said implied anything different. Fuck off.

>>12414845
There are four cases where one coin is known to be heads.

>>12414877
>There's no "the other one". you don't know which coin is heads.
Non sequitur. You don't have to know which is which, we're told one is heads.

>>12414907
We're not told which one is heads. That's why the probability is 1/3

>>12414899
Flip two coins, what can happen?

HH 1/4
HT 1/4
TH 1/4
TT 1/4

We are told at least one is H. We go back
To our probabilities.

HH/(HH+HT+TH) = (1/4)/(3/4) = 1/3

Hoooooly shiiiit I’m smarter than scitards!

>>12414899
>a 50% chance the known heads
You keep pretending you get to "know" which coin that is. You don't, you're just making it up.

This is conditional probability P(A | B) = P(A intersect B)/P(B)

P(A intersect B) = P(A | B) * P(B)
P(B) = P(A | B) = 1/2

Therefore it's 1/4

>>12414883
If you flip two coins HH can only happen in one way.
Correct. But if you flip two coins and then observe one coin is heads This happens in two ways.

>If you don’t know which coin is H the probability is 1/3.
Incorrect. For example, if your friend Bob told you he saw a coin and it was heads, then the answer is 1/2, even though you don't know which coin Bob saw. Again, it doesn't matter which coin is the known coin. By symmetry its the same.

>>12414877
Exactly, this poster is trying to cheat by making up a step where you "find out" which coin is "known" and which coin is "unknown", and then use this fake information to create a new coin flip out of the "unknown" coin.

>>12414917
>We're not told which one is heads
See >>12414907

>>12414948
>and then observe one coin is heads
You keep pretending you get to "observe" the "known" coin. You don't, you're just making it up.

>>12414952
Your post makes no sense. What's non sequitur? You don't know which coin is heads, do you agree with this?

>>12414948
>if your friend Bob told you he saw a coin and it was heads
First, this is just fake information you're making up, so who cares.

Second, even if you add this fake information, it will only "pin down" the "known" coin if he told you which coin he saw.
If he didn't tell you "which" coin he "knows" you didn't even learn anything new, so nothing changes and the answer is still 100%.

>>12414919
>We are told at least one is H.
Yeah, so there is a 50% chance that either coin is that coin:

Hh
hH
Ht
tH

The fact that it's twice as likely to observe one heads when there are two heads increases the chance that there are two heads.

>>12414948
>If you flip two coins HH can only happen in one way.
Then you have conceded that the following is true:

HH 1/4
HT 1/4
TH 1/4
TT 1/4

Then how can this be anything but 1/3? In order for it to be 1/2 you need to exclude either HT or TH. In order to exclude HT or TH you need to know which coin is heads, which you don’t, you are only told that one of the coins is heads.

>>12414940
>You keep pretending you get to "know" which coin that is.
No, in fact I explicitly said we don't know which coin it is. Learn English.

>>12415095
>Hh
>hH
These are the same events. You're answering a different question than what's being asked

>>12414951
>Exactly, this poster is trying to cheat by making up a step where you "find out" which coin is "known" and which coin is "unknown"
No I'm not. I explicitly said we don't know which is which, and it doesn't matter. Doesn't matter how many times you repeat this lie, my posts speak for themselves.

>>12414998
>What's non sequitur?
Knowing which coin is which is a non sequitur.

>You don't know which coin is heads, do you agree with this?
Yes.

People here actually have the audacity to say /sci/ is a high IQ board.
I'm not even shitting on the 50%fags or 0.999... =/=1 trolls, its you retards who keep replying when they're obviously moking you.

This shit is getting 200+ replies, nuke this board.

>>12415104
It does matter whether you know which coin is heads, or whether you don't know which coin is heads.

>>12414999
>First, this is just fake information you're making up, so who cares.
It's called an EXAMPLE. Ever heard of that word? It shows that it doesn't matter whether you know which coin is heads as long as you know a coin is heads. This refutes your claim that if you don't know which coin is heads the answer is 1/3.

>Second, even if you add this fake information, it will only "pin down" the "known" coin if he told you which coin he saw.
Meaning what?

>If he didn't tell you "which" coin he "knows" you didn't even learn anything new, so nothing changes and the answer is still 100%.
You just said the answer is 1/3. Make up your mind. And yes, my point is that nothing changes, so the answer remains 1/2.

>>12415107
>Yes
Then TT is the only possibility excluded from the four HH, TH, HT, TT

>>12415103
>These are the same events.
No, they're not. A coin cannot be both known to be heads and not known to be heads.

>>12415096
>Then you have conceded that the following is true:
Yes, that's true before you're told a coin is heads. Good job.

>Then how can this be anything but 1/3?
I already told you.

>In order for it to be 1/2 you need to exclude either HT or TH.
If we order the events such that the first coin in our order is the coin known to be heads then we get:

HH
HT

1/2

>>12415120
>your claim that if you don't know which coin is heads the answer is 1/3.
What?
>You just said the answer is 1/3.
What? I've been saying it's 100% since yesterday.

Stop trying to weasel out of your lies.

>>12415130
>the first coin in our order is the coin known
There's no "the coin known" you retard

>>12415104
>my posts speak for themselves
Yes they do. All of your posts involve making up fake information about a "known" coin, and then pretending that you didn't

>>12415137
Exactly, as I said here
>>12414951
this poster keeps trying to cheat by pretending they (or their friend Bob) gets to "observe" a coin and pin it down as the "known" coin.

Once you see the trick, it's easy to point it out every time.

>>12415110
>I'm not even shitting on the 50%fags or 0.999... =/=1 trolls, its you retards who keep replying when they're obviously moking you.
I am mocking you, but not for the reason you think.

>>12415122
>Then TT is the only possibility excluded from the four HH, TH, HT, TT
Wrong. See >>12415130

>>12415171
>Wrong
you haven't explained why is this reasoning wrong

>>12415130
>If we order the events such that the first coin in our order is the coin known to be heads

>>12415195
They can't. They're just trying to cheat by making up fake information, and hoping you won't notice the trick.

The trick is they pretend
>at least one is heads
is the same thing as
>at least one is heads
>you know which one it is

>>12415131
You're replying to a post replying to >>12414883 which says the answer is 1/3. And your explanation of 100% is complete gibberish.

>>12415257
>You're replying to a post replying to
There are more than two people on this board, and your explanation of 50% is both lying and gibberish

>>12415231
>the coin known to be heads
Yep, Every time, ha ha.

>>12415137
>There's no "the coin known" you retard
"At least one coin is heads." Learn how to read.

>>12415140
What information did I make up? You don't even understand what a probability is.

>>12415152
>this poster keeps trying to cheat by pretending they (or their friend Bob) gets to "observe" a coin and pin it down as the "known" coin.
Where did I say I know which coin is heads? Just show it once and you win.

>>12415303
>"At least one coin is heads."
exactly. and yet this doesn't imply there's a "known coin".

>>12415195
I did several times. You're ignoring that a coin is known to be heads.

>>12415312
no, there's not a coin known to be heads. the only thing that's known is that TT haven't occured. learn how to read.

>>12415231
>>12415278
Not an argument, try again.

>>12415249
>>you know which one it is
I never said or implied that. Stop lying.

>>12415310
That's exactly what it implies.

>>12415324
>>12415318
>>12415309
>>12415305
>Where did I say I know which coin is heads?
All of your posts lie about that. Here's just one example

>>12415130
>If we order the events such that the first coin in our order is the coin known to be heads then we get:
>
>HH
>HT
>
>1/2

>>12415318
>i don’t know which coin is heads
>let’s say coin A is heads

>>12415332
You just lied about it again.

>>12415317
>no, there's not a coin known to be heads.
"At least one coin is heads"

>the only thing that's known is that TT haven't occured.
Right, but you're assuming all other permutations are equally likely given that information. That's false since you're twice as likely to observe one heads if it was HH. So HH is twice as likely as TH or HT.

>>12415355
>you're assuming all other permutations are equally likely given that information
yes, that's how it works
>That's false since you're twice as likely to observe one heads if it was HH. So HH is twice as likely as TH or HT.
complete gibberish

Step A (real information)
"you flip two coins"
>1. (H)(H)
>2. (H)(T)
>3. (T)(H)
>4. (T)(T)

Step B (real information)
"at least one is heads"
>1. (H)(H)
>2. (H)(T)
>3. (T)(H)

Step C (fake information)
* * * * * you know which one is it * * * * *
>1. (H)(H)
>2. (H)(T)
or
>1. (H)(H)
>3. (T)(H)

>>12415333
>Here's just one example
This doesn't say that we know which coin is heads, only that we *ordered* the coins according to whichever one is known to be heads. This is the same as

Hh
hH
Ht
tH

Except we just shifted the capital letters so that the known coin in each case is first:

Hh
Hh
Ht
Ht

Which is the same as

HH
HT

>>12415355
>as likely to observe
You have a 0% of observing anything.
Notice this is always the trick.
"Observe"
"Known coin"

>>12415340
>>let’s say coin A is heads
Who are you quoting?

>>12415387
>according to whichever one is known to be heads
look, he did it again

>>12415387
>whichever one is known to be heads
Fake information, you're just making it up.

>>12415343
No I didn't. >>12415363
>yes, that's how it works
No it isn't.

>complete gibberish

>>12408192
50% since the first coin throw is already predetermined and doesnt matter

>>12415390
>>let’s say coin A is heads
>Who are you quoting?
>>12415387
>whichever one is known to be heads

>>12415363
>yes, that's how it works
No it isn't.

>complete gibberish
It's not. Take a basic probability course. Here's an example:

You have three boxes. Box 1 has two gold coins. Box 2 has a gold coin and a silver coin. Box 3 has two silver coins. You pick a box at random and a coin from that box at random. It's gold. So what is the chance you chose from Box 1? Ah, according to your logic it must be 1/2 since Box 3 is eliminated as an option and the Box 1 and 2 were equally likely. But that answer is wrong. Since there are two gold coins in Box 1, it's twice as likely we chose from Box 1. The answer is 2/3.

>>12415398
>No it isn't.
yes, it is. excluding an event doesn't magically change the relative likelihoods of the remaining events. all you know from the problem is that an event hasn't occured so it must be excluded.

here's a problem for you: you throw two dice. the sum of both is 6. what's the probability that both are 3?

>>12415374
>Step B (real information)
>"at least one is heads"
>>1. (H)(H)
>>2. (H)(T)
>>3. (T)(H)
Right, but HH is twice as likely as HT. You're exhibiting the freshman mistake of assuming events are equally likely just because you listed them.

>>12415395
>>12415396
Not information, just arbitrary ordering.

AB can be the same as BA, if you're just using a different ordering.

>>12415435
>Right, but HH is twice as likely as HT.
no, there's absolutely no information which indicates that HH happens twice as likely than HT or TH

>>12415411
In that example, you're picking a box and observing a coin.
This is just more of your fake information you keep trying to make up.
All you ever do is try to make up fake information, sneak it in, and lie about it whenever you get caught, which is in every post you make.

In this example, you're not picking anything or observing anything.

>>12415405
>>whichever one is known to be heads
Yeah, that could be coin A or coin B. I didn't say we know which is which, only ordered according to it. For example, if I have coin A and B and one is 5 cents and one is 10 cents, but I don't which then the possible permutations are

5A 10B
10B 5A

Or I could order it according to whichever is 10 cents:

10B 5A
10A 5B

If I did that would you suddenly say I'm claiming I know which is 10 cents? No. But that's what you've done throughout this thread.

>>12415411
>>12415411
that's a completely different problem. "Box 3" is not elementary event, but TT is.

>>12415435
>HH is twice as likely as HT.
Only when you make up fake information and lie about it, liar.

>>12415465
>only ordered according to it
If you don't know something, you can't use it to order by.

>>12415463
>In that example, you're picking a box and observing a coin.
Distinction without a difference. If your friend Bob did the same and told you only that at least one coin in the box was gold, would you change your answer to 1/2 because you didn't choose a box or observe a coin? No.

>This is just more of your fake information you keep trying to make up.
What information? The only information used was that at least one coin in the box is gold. The same information given in the problem.

>>12415411
Right, you solve that problem in the same way you solve this problem.

G1
G1
G2
S2
S3
S3

All are at 1/6. If you pick up a gold coin you exclude all silver coins and it’s 2/3 chance from box A since:

G1 + G1 = 1/6 + 1/6 = 2/6

And (G1+G1)/(G1+G1+G2) = (2/6)/(3/6)=2/3

>>12415465
>if I have coin A and B and one is 5 cents and one is 10 cents
More fake information you're trying to sneak in, with a fake analogy.

We don't know that one coin is "known" and one is "unknown", all we know is that at least one coin is heads.

>>12415483
>your friend Bob did the same
More fake information, no one "observes" anything. Not you, not your friend Bob.
Nothing to do with the problem. All lies and fake.

>The same information given in the problem.
No, you're choosing one of the boxes. In the problem, you're not choosing one of the coins.

>>12415467
>that's a completely different problem.
It's a different problem, yes. It's an example of how your logic fails.

>"Box 3" is not elementary event, but TT is.
And?

>>12415478
>If you don't know something, you can't use it to order by.
We know a coin is heads, so it's fine to order by it. The order is arbitrary anyway, so you're not even making a relevant point. Order it however you want, you get the same answer.

>>12415470
See >>12415483

>>12415506
>>12415513
>>12415519
As you can see from this post the correct way of solving your problem is the same as solving this problem (discard

>>12415488

You have effectively proven me right. The answer to this problem is 1/3 and to your problem 2/3 and both can easily be solved using the correct method.

The problem of this thread:

HH (1/4)
HT (1/4)
TH (1/4)
TT (1/4)

Solution: discard TT and solve HH/(HH+HT+TH)

Your problem

G1 (1/6)
G1 (1/6)
G2 (1/6)
S2 (1/6)
S3 (1/6)
S3 (1/6)

Solution: discard silver coin and solve (G1+G1)/(G1+G1+G2)

Brilliant

>>12415506
And?
if you exclude an ELEMENTARY event, the relative probabilities of the remaining ELEMENTARY events are unchanged. that's conditional probability. this logic leads to the correct answer 2/3 of your problem.
obviously it doesn't work for non-elementary events, therefore no, "by my logic" eliminating Box 3 doesn't mean Box 1 and 2 are equally likely.

>>12415496
>More fake information you're trying to sneak in, with a fake analogy.
How is it fake? You have no argument so you just cry "fake" over and over again.

>>12415496
>We don't know that one coin is "known" and one is "unknown", all we know is that at least one coin is heads.
That's what at least one coin is heads means.

>>12415506
>>12415555
I'll add that by "relative probabilities" I mean the ratios P(A) / P(B), i.e. how much is one event more likely to happen than another event. it's all conditional probability, excluding an event X is just passage to a new probability function

new probability of A = P(X | not X)

which in the case of elementary events is just dividing by P(not X)

>>12415573
>P(X | not X)
P(A | not X)

>>12415558
>That's what at least one coin is heads means.
No, it doesn't mean that one coin is "known" and the other coin is "unknown"
That's the fake information you keep making up.

>>12415558
you just cry "fake" over and over again
You just keep making up the same fake information over and over again, what do you expect?

>>12415562
>That's what at least one coin is heads means.
No, it doesn't mean that one coin is "known" and the other coin is "unknown"
That's the fake information you keep making up.

>>12415558
>you just cry "fake" over and over again
You just keep making up the same fake information over and over again, what do you expect?

Literally a textbook problem. It is 1/3.

Got something for you schizo.
According to your wikipedia article, the question is ambiguous because no information is given on how your imaginary friend came to the conclusion that "at least one is HEADS".
Let's assume this is totally not an autistic take on a simple problem.
I will make it a point to follow this line of logic here.

So we don't know how we got "at least one is HEADS" right?
What if only the first coin is known, and it turns out to be heads?
Then it satisfies the condition.
However the probabilities change and this ultimately leads to a probability of 1/2. Ok.

But that would be assuming only one coin was known to make that assessment, right?
But we don't know that.
Since we don't have information, we don't know if either one or both coins were known.
We therefore have to take this into account as well.

Let T/t be tails, H/t be heads, where uppercase letter means the coin is known and lowercase means the coin is unknown.
To satisfy the condition "at least one is heads", one coin must be heads AND known (uppercase H).
Here are every 12 possible events, considering 4 arrangements, independant coins and the possibility of 1 or 2 known coins.

Tt
Th
Ht
Hh
tT
tH
hT
hH
TT
TH
HT
HH

Out of these 12 events, 7 satisfy the condition "at least one is heads". 3 out of these 7 are events where both coins are heads.
Given at least one is heads, the probability that both are heads is 3/7.
So even following your line of logic, your retard 1/2 answer is blown out of the water. I'm still not done here though.
3/7.

>>12415717
kek

>Hh and hH are the same scenario you imbecile
>NOOOO TH AND HT ARE DIFFERENT SCENARIOS NOOOO IT CANT BE 1/2

Fucking retards it is either 1/2 or 1/3 depending if both coins were flipped at the same time, if that is the case then it is 1/3, if the the first coin was flipped, then the second was flipped after, it is 1/2

>>12415504
>More fake information, no one "observes" anything.
It's an example.

>Nothing to do with the problem.
Wrong.

>No, you're choosing one of the boxes. In the problem, you're not choosing one of the coins.
So?

>>12415740
TH and HT can be considered to be the same. but then TH=HT has different probability than HH or TT.

>>12415717
>>12415734
>>12415740
>>12415761
It’s a textbook problem. I have the answer, it’s 1/3. You can stop arguing like retards now.

>>12415761
No it is 1/2 regardless of if they were flipped at the same time or one after the other. One coin will always be heads, the only coin on which probability lies is the second coin

>>12415488
Right, and you solve the original problem this way:

H
T

1/2

>>12415790
>the only coin on which probability lies is the second coin
You don't and can't know which coin is the "second" coin.
More fake information, made up lies.

>>12415790
>the second coin
look, he did it again

>>12415793
More cheating with fake information.

The trick is to lie and pretend that
>at least one is heads
gives the same result as
>at least one is heads
>you can know which one it is

>>12415793
No it’s 1/3. 1/2 is only possible if you know which coin is heads.

>>12415801
>>12415808
the "other coin" then, defined as the one that was not predetermined, if thats what makes you happy.
the only options are:

_H in which the "other" coin in the blank space has a 1/2 chance of being heads
and
H_ in which the same as above is true.

>>12415846
There is no predetermined coin. Two coins are tossed and at least one is heads.

>>12415846
>the one
look, he did it again

>>12415851
Yes one coin is predetermined to be heads since one of the coins MUST be heads

>>12415846
>the "other coin"
>the second coin"
>the unknown coin
It doesn't matter what "word" you use to mark a coin with your fake information, the point is that you can't mark a coin. Completely fake.

>the one that was not predetermined
No "one" coin is "predetermined", this is just the same made up nonsense over and over again. Fake information.

>>12415860
what are you implying

>>12415717
(2/2)
However, we get 3/7 by assuming every single possible event follow a flat distribution.
That would mean the probability of knowing only one coin is twice the probability of knowing both coins.
What if the probability of knowing only one coin is EQUAL to the probability of knowing both coins? (aka flat distribution of the number of coins known)

Therefore both cases would share 50% of the total set of events.
1 known coin -> 8 arrangements.
2 known coins -> 4 arrangements


Tt * 1/16
Th * 1/16
Ht * 1/16
Hh * 1/16
tT * 1/16
tH * 1/16
hT * 1/16
hH * 1/16
TT * 2/16
TH * 2/16
HT * 2/16
HH * 2/16

At least one is heads:

Ht * 1/16
Hh * 1/16
tH * 1/16
hH * 1/16
TH * 2/16
HT * 2/16
HH * 2/16
10/16

Both coins are heads:
Hh * 1/16
hH * 1/16
HH * 2/16
4/16

Probability both coins are heads given one coin is heads: 4/16 / 10/16 = 2/5.
So assuming another proportion of 1 known coins vs 2 known coins, we get 2/5.
So what is is?
1/2?
3/7?
2/5?

The thing here is we don't know this proportion, we don't know the probability of knowing only one coin, or both.
No information is given, at all, and no assumption can be made.
You can't just say "at least one is heads" is the result of knowing only one coin. You don't know that.
You would have to consider all possibilities.
However, you don't know the probability of either number of known coins, and can't assume a flat distribution.
Therefore, there is no one single answer.
That would render the problem incomplete, with an infinite number of solutions.
That is following your line of logic that you need to know how the information was obtained.

Or you can just interpret this problem like every other textbook problem and let the information given by the predicate become the hypothesis which sends you on the path to one single answer.
Which is 1/3.

>>12415867
The result of flipping two coins is at least one heads. This has nothing to do with any specific coin being "predetermined", that's your fake information you keep trying to sneak in.
We're on to your trick, though.
No matter how many different words you use to tell the same lie, it's not going to suddenly come true.

>>12415867
Do you know what predetermined means? It means that the outcome is decided before the toss is made. In OP it is clear that the toss was made and the outcome was that at least one coin is heads.

>>12415776
Exactly. The HT/TH event comes up 50% of the time.

>>12415880
that you're making the same mistake over and over again. you first assume that result of a specific coin is known, solve it to get 1/2, and then you think you can get rid of the auxiliary assumption by saying that the cases are symmetric and equally likely to happen, so the probability is

1/2*1/2 + 1/2*1/2 = 1/2.

that's a solution to a different problem.

>>12415871
>No "one" coin is "predetermined"
>at least one coin is heads

>>12415761
Doesn't change anything.
If the coins are flipped at the same time, they are still independant.

>>12415902
true. information in OP then says to discard TT. that leaves HH with 1/3 prob and HT with 2/3.

>>12415908
at least one coins can refer to either of them, or both, as long as it satisfies X >= 1

>>12415902
>The HT/TH event comes up 50% of the time
If the TT results are included. But the question excludes them. So they come up 2/3 of the time in the included results.

>>12415781
We all know it's 1/3.
It is a standard textbook problem.
I am trying to show schizo here that is line of logic is faulty and that the 1/2 answer is retarded

>>12415896
It is decided in advance that one of the coins in the toss will be heads. i'm not claiming to know which coin that is or trying to "mark" any specific coin, i am only speaking that both before and after the outcome is observed, one of the coins WILL be heads. There is only an option present for one coin

>>12415908
Correct.

>>12415923
Of course. I meant 50% of the totral set of events.

>>12415921
I agree
so we are left with the following possible scenarios
H_ (1/2 chance)
or
_H (1/2 chance)

>>12415717
>>12415885
/thread

>>12415953
that would be assuming you only know one coin.
But how do you know the number of coins known?

>>12415933
>observed
Sorry, you still don't get to "observe" anything.
>There is only an option present for one coin
No, both coins have the same option, they either give a heads result or a tails result.

All you know beyond that is that, given two flips at least one result of the two flips is heads.
You can't tie that to "one coin" that you then magically follow back in time to make a "predetermination" with.

>>12415953
>we randomly select a coin and declare it to be heads. then we flip the other coin. what's the probability that both are heads?
this is the problem you're solving

>>12415953
I want to see you solve >>12415417

Threads like this are autism magnets to keep you retards out of other threads

>>12416043
No way, this thread is fucking hilarious.
The same hallucinating schizo I ass raped into oblivion, by simply citing the exercises and examples in the pdf he linked, is now literally sock puppeting his own schizo "muh bOy grLL" hallucination to try and convince himself it makes sense.
And getting absolutely ass raped in the comments, yet again, by all comers.

>>12415953
As if any of this retarded garbage this is helping your cause in the least, you fucking worm brained illiterate schizo.
And fuck your imaginary friend "Bob," too, LMFAO

>>12416019
the answer to that is 1/5

>>12416005
i'm solving the question in the OP, not the stawman you want me to be solving

>>12416090
and solution?

>>12415540
>The problem of this thread:

>HH (1/4)
>HT (1/4)
>TH (1/4)
>TT (1/4)
Right.

>Solution: discard TT and solve HH/(HH+HT+TH)
Correct, but the fact that at least one coin is heads makes HH twice as likely to have occurred.

HH (1/2)
HT (1/4)
TH (1/4)

>>12408192
A good way to see what's happening is this:

Get 2 coins and toss them simultaneously. If they are both tails then record nothing and go again. Otherwise record if it's 1 head or 2. Repeat until you have 100 results (of 1 head or 2).

Now do another experiment but this time toss coins one at a time. If the first is tails record nothing and go again, otherwise flip the second and record the result. Repeat until you flip the second coin 100 times.

Now it should be obvious the results will be different.

>>12416168
see >>12415957

>>12416138
favourable outcomes/possible outcomes
i.e.
5 ways the dice can add to 6 but only in one of them are both dice 3

>>12415957
one coin is known to be heads for sure, so 1 coin is known

>>12416168
>Correct, but the fact that at least one coin is heads makes HH twice as likely to have occurred.
>HH (1/2)
>HT (1/4)
>TH (1/4)
Not in this case, all three satisfy the statement ”at least one coin is heads” equally.

>>12415555
>if you exclude an ELEMENTARY event, the relative probabilities of the remaining ELEMENTARY events are unchanged.
If that's true then HH is not an elementary event in this problem just as Box 1 is not an elementary event. Rather the elementary events are

Hh
hH
Ht
tH
tt

Ignoring that it becomes known that one coin is heads when setting up the sample space is probably where your issue lies.

>>12415593
>No, it doesn't mean that one coin is "known" and the other coin is "unknown"
It does. "At least one coin is heads."

>That's the fake information you keep making up.
Information given directly by the problem can hardly be fake.

>>12416180

Possible outcomes (sums)

2 1/36
3 2/36
4 3/36
5 4/36
6 5/36
7 6/36
8 5/36
9 4/36
10 3/36
11 2/36
12 1/36

Discard everything but 6 and you are left with 5/36. Possible outcomes:

1+5 1/36
2+4 1/36
3+3 1/36
4+2 1/36
5+1 1/36

Probability for 3+3 is then:

(1/36)/(1/36+1/36+1/36+1/36+1/36) = (1/36)/(5/36) = 1/5

This is a long solution I just wrote it down to show the same method is valid for any of these kinds of problems in the thread.

>>12416209
>It does. "At least one coin is heads."
Nope, see>>12415990
You're pretending that a result of two coin flips can be tied to one coin.
Totally fake gibberish.

>Information given directly by the problem can hardly be fake.
I know, that's why you keep having to make up your own fake information that isn't given directly by the problem.

>>12415717
>>12415885
>No information is given, at all, and no assumption can be made.
Yes, that has been my point throughout the thread. You finally got it.

>Or you can just interpret this problem like every other textbook problem
I'm not aware of other textbook problems like this. You don't seem to be aware that what you consider the "default interpretation" or the "simplest answer" is merely another arbitrary assumption based on nothing. This is why psychologists study this problem. You are so invested in your own naive heuristic that you can't see past it.

>>12415776
Hh and hH can be considered to be the same. but then HH has different probability than TH or HT.

>>12416246
>You are so invested in
Lmfao, as predicted >>12416079
The schizo reveals himself.
You're so invested in your schizo wikipedia hallucination that doesn't correspond to the actual OP, that you just sock puppeted you own horse shit for a day and still couldn't sell it to anyone.
AHAHAHAHAHAHA

>>12415818
Done with your tantrum yet?

>>12416263
Not me, but are you done masturbating to your schizo "psychologist" hallucination yet? Lmfao, absolute retard.
Let me know when you publish the results of all this on vixra, you illiterate fuck.

>>12415836
>No it’s 1/3.
Why?

>1/2 is only possible if you know which coin is heads.
Nope, it makes no difference. Let uppercase H be the known heads. Even if we don't know which coin it is, the possible permutations are:

Hh
hH
Ht
tH

So it's 1/2

>>12416292
If you don't know what coin it is, Hh and hH are the same result.

>>12416228
Possible outcomes for the OP:
HH
HT
TH
TT
Discarding those that dont fit the criteria you are left with:
HH
HT
TH
TH and HT are the same, since there is no "first" and "second" coin, therefore:
HH - set which contains 2 heads (1/2)
HT/TH - set which contains heads and a tails (1/2)

>>12408192
I think the ambiguity comes from the implications of the statement "at least one is heads". There are really two different scenarios that could be assumed.

Let's say someone flipped two coins and covered them up with their hand. He then moves his hand revealing one coin to be heads. In this case the probability that both coins are heads, is the probability that the hidden coin is heads which is simply 1/2. More specifically, what has happened here is that two states have been removed from possibility: TT and TH. Now only HT and HH remain and are both equal probability so that correct answer is 1/2.

On the other hand, if the man flips two coins, covers them with his hand, and tells you at least one of these coins is heads. In this case only the state TT has been removed. The states TH, HT, and HH remain, so the correct answer would have to be 1/3.

>>12416300
>HH - set which contains 2 heads (1/3)
>HT/TH - set which contains heads and a tails (2/3)
ftfy

>>12416295
HT and TH are also the same result

>>12415860
You know there are multiple people in the thread right?

>>12416308
if HT and TH are separate then Hh and hH are separate as well, bringing us back to 1/2

>>12416302
>He then moves his hand revealing one coin to be heads.
Exactly, this is the fake information this poster keeps making up and trying to sneak in.
We're all on to the trick

>>12416322
>>12416309
Because why? Magic? Your trick will never pass, everyone sees what you're doing.

>FAKE FAKE FAKE INFORMATION FAAAAKE

>>12411840
We don't know WHICH coin is heads dumbass. Only that at least one of them will be.

>>12408570
It is 1/3 that anon >>12408195 is just a brainlet

Please stop arguing and read >>12416173 it answers all your problems. You can even do the experiment to convince yourself!

>>12416343
Don't get mad, all you have to do is stop making up fake information and I'll stop pointing it out.

>>12416333
Why is hH and Hh the same result but not TH and HT? Because you say so? They both have to do with positioning of the coins, but as we sais before there is no markin of coins and no "first" and "second" coin. You can't apply your imaginary rules for one set but not the other, schizo

>>12416079
You completely failed to argue against 1/2. You're claiming victory again even though you already lost.

>Gardner initially gave the answers 1/2 and 1/3, respectively, but later acknowledged that the second question was ambiguous.[3]
>Grinstead and Snell argue that the question is ambiguous in much the same way Gardner did.[12]
>From the position of statistical analysis the relevant question is often ambiguous and as such there is no “correct” answer.

>>12416300
>TH and HT are the same, since there is no "first" and "second" coin, therefore:
>HH - set which contains 2 heads (1/2)
>HT/TH - set which contains heads and a tails (1/2)
non sequitur. HT=TH has a larger probability of happening than HH. if you claim that there's 50/50 chance between TH=HT and HH, then answer to this problem:

"you toss two coins. what's the probability that both are heads."

is 1/3. that's obviously wrong.

>>12416374
>>12416373
>>12416358
>>12416355

Please do
>>12416354
irl

>>12416358
You're trying to sneak a lower/upper case option into only one result of your permutation. If that's an "option", then you get 6 equally likely "outcomes", not 4.
hH Hh tH Th hT Ht

>>12416183
But maybe two coins are known and only one of them is heads.
How do you known how many coins are known?

>>12416175
The problem states one coin is known. Assuming the other is known is assuming more information than you have. Fake information if you will.

>>12416373
Go suck off your cellmate Bob again, schizo.
LMFAO

>>12416246
>Yes, that has been my point throughout the thread. You finally got it.
I made that point following your line of logic.
I showed this leads to a dead end.
Read the whole post.

>>12416185
>Not in this case, all three satisfy the statement ”at least one coin is heads” equally.
Non sequitur. You are taking the ratio of their probabilities, not their "satisfaction."

>>12416373
Oops, you dropped it again
>...depending on what information was available beyond that just one child was a boy.

>>12416309
Nope. Permutations matter.
Hh and hH are considered different if you assume numbers of known coins, which I proved to be retarded

>>12416317
Yes, one schizo and other anons

>>12416300
HT means coin A is heads coin B is tail
TH means coin A is tail coin B is heads
HH means coin A is heads coin B is heads and can only happen in one way if you toss two coins.

>>12416423
1/2 prob of satisfaction by HH when only one coin is known,
But, >>12416404

>>12415417
>excluding an event doesn't magically change the relative likelihoods of the remaining events.
Right, but "at least one is heads" doesn't just exclude an event.

>here's a problem for you: you throw two dice. the sum of both is 6. what's the probability that both are 3?
1/5

>>12416322
Nope. The first is permutations of outcomes, second is permutations of satisfaction with your assumed "only one coin is known"
The two are not equivalent and one does not imply the other

>>12415443
>no, there's absolutely no information which indicates that HH happens twice as likely than HT or TH
"At least one is heads"

>>12416423
If if’s HT Bob will say ”at least one is heads”

If its TH Bob will say ”at least one is heads”

If its HH Bob will say ”at least one is heads”

Listen to Bob, stop, stop taking drugs, stop.

>>12416428
What matters is the probability when the two coins were tossed and what outcome can be excluded with Bob’s help.

>>12416407
>The problem states one coin is known.
Where?
"at least one is heads" doesn't provide the number of known coins.
You could make the same assessment by knowing both coins.

>>12416290
So that's a no.

>>12416466
who the fuck is bob?
since when is there a bob in the problem?

Oh ok so you guys are deliberately ignoring solutions that would stop your shitposting? You can literally toss coins irl to see the probability, there is no argument.

Carry on schizos

>>12416397
h refers to a heads coin that is landed after the condition of "at least one heads" has been satisfied therefore Th and hT are same as TH and HT, it is only necessary in sets already containing H to differentiate

>>12416481
We all did this a long time ago, also proven by independant anons with simulation.
But one schizo likes to play with words

>>12416482
Since when is the condition verified between two flips?
That is still assuming you only verify one coin which is autistic.

>>12416454
>"at least one is heads" doesn't just exclude an event.
of course it does

>>12416246
>psychologists
This is actually a fascinating glimpse into the method and mechanisms of psychology itself.

A schizo hallucinates random extraneous information into a set of words that don't convey that information, then tries to convince everyone else that his hallucination really does exist in those words.

It's all variations on "my jacket button is slippery" = "I want to lick my mother's nipples" or whatever the tripe du jour is.

Pure, unadulterated horse shit, which has nothing to do with the simple, unambiguous question in the OP.

Damn, can't tell if people are baiting or actually overthinking such a simple problem. This is a question of combinations, not permutations. At least one coin is heads. HT-HH-TH-HH are the possible outcomes, if you want to needlessly count the permutations. 2/4 outcomes are both heads. This is supposed to be the smart board...

>>12416478
Bob is our best friend. Trust Bob. Listen to Bob.

>>12416521
>This is a question of combinations
>HT-HH-TH-HH
Ah yes, HH and HH, the two distinct combinations

>>12416521
>HT-HH-TH-HH
you got HH twice

>>12416521
Please come here. Bob wants to have a word with you.

>>12416482
>landed after the condition of "at least one heads" has been satisfied
More fake information. It's two flips, then the condition. Not one flip, the condition, then the second flip.
We're on to your tricks.

>>12416241
>You're pretending that a result of two coin flips can be tied to one coin.
"At least one is heads" is one coin.

EVERYONE LISTEN TO BOB!

>>12416544
No, it's a partial result of two coins.

>>12416528
kek, tell that to schizo

>>12416544
You keep avoiding the question.
see >>12416467

>>12416534
>>12416530
Yes, counting both permutations HH does in fact appear twice, two distinct events. If you insist on counting HH only once, you must also count TH and HT as one event, since the order of the coins is not relevant.

>>12416557
kinda this, yeah.
and if we were to consider the number of coins that are verified, we can get either 1/2, 3/7, 2,5 and an infinity of other retard answers.

>>12416576
>HT-HH-TH-HH are the possible outcomes, if you want to needlessly count the permutations
I even included the statement that I was counting all permutations and you geniuses elided right over it... you either took too many math classes or not enough

>>12416295
>If you don't know what coin it is, Hh and hH are the same result.
Doesn't matter whether you consider them the same result.

>>12416576
>HH does in fact appear twice, two distinct events.
how are the events distinct

>>12416576
There are no two permutations of HH. HT and TH are different events because the coins are independant.

HT:
Coin A = H
Coin B = T

TH:
Coin A = T
Coin B = H


HH:
Coin A = H
Coin B = H
This is the only permutation of HH.


Just look up what permuations actually mean instead of talking about something you don't know.

>>12416594
I know, it comes to 1/3 with or without your trivial upper/lower case game.

>>12416544
no, it's an information about two coins which cannot be reduced to an information about a single coin.

>>12416589
you either took too many math classes or not enough
apparently you took none since you can't make the difference between a permutation and a combination

>>12416612
Also well said, exactly.

>>12416506
Not an argument.

>>12416302
>On the other hand, if the man flips two coins, covers them with his hand, and tells you at least one of these coins is heads.
This is still ambiguous how did he determine there was at least one heads?

>>12416344
>We don't know WHICH coin is heads dumbass
He didn't say we do, nor does it matter. Learn how to read.

>>12416684
{TH,HT,HH} = {TT,HT,TH,HH} - {TT}

>>12416693
>determine
No one "determined" anything. You have the information you have and that's it.

>>12416693
>he
who the fuck is he?
there is no Bob, no mr.smith

Again, you decide you need to consider how this information was "obtained", you open the question to an infinite number of solutions, which renders the problem invalid.

There
's

no
rawndom
in tossing
a
coin/

>>12416481
show me one correct solution that was posted, i'll wait

>>12416425
>>...depending on what information was available beyond that just one child was a boy.
What information is available beyond that? Are you saying that it's unambiguous as written by adding more information makes it ambiguous? LMAO. Nice cope.

>>12416397
Th and hT get removed once it's revealed that a coin is heads. You still don't get it.

>>12416404
>But maybe two coins are known
Fake information. Only one coin is known to be known.

>>12416419
>I made that point following your line of logic.
Yes, and your only "argument" against it was "I'm just going to assume it only has one interpretation even though I already admitted it's ambiguous."

>>12417124
>Only one coin is known to be known.
see >>12416467

>>12416461
Yes all of that is true. What is your point? It's twice as likely HH occurred if Bob says "at least one is heads."

>>12417128
Learn to read.
I didn't say it was ambiguous.
I showed that if you consider the question to be ambiguous because you need to know how the information was obtained, you end up with infinite solutions and the answer is indeterminate.

>>12416428
You assume two coins are known, i.e. it would always be reported that one is heads if the pair contains heads. I guess you're retarded.

>>12417132
If you only verify one coin.
But where does the "know only one" come from?

>>12417139
Yes, that is the meaning of "at least one is heads".
One in the pair is heads.

>>12417128
nothing about OP is ambiguous
>>12417124
no coins are ever “known to be known”>>12417121
nothing is ever “revealed”

All you do is keep making up fake “observations” and “reveals”, it’s all fake information, everyone knows it, you’re not fooling anyone

>>12416467
>Where?
"At least one coin is heads"

>"at least one is heads" doesn't provide the number of known coins.
So it's ambiguous? No I don't think so. Any sane person would see it only tells you about one coin. I'm sane. I'm a stable genius. Everything I immediately believe must be true.

>>12416481
Flip a coin and see it comes up heads half the time if you don't believe me.

>>12417139
>You assume two coins are known
there are no "known" or "unknown" coins. how do you even solve the problem with dice with logic? if you know that sum is 6, that clearly means both dice are known. how come the probability is not 1? or 0?

>>12416557
Yeah, that partial result being one of the coins is heads. Assuming you know something about both coins is just fake information hallucinate by a schizo.

>>12416570
Patience, little crybaby.

>>12417155
You can make the same assessment by seeing two coins.
You see both coins. One is heads and the other is tails. At least one coin is heads.
You see both coins. Both coins are heads. At least one coin is heads.
X = 1 X ≥ 1
X = 2 X ≥ 1

>>12416608
Then why are you talking about same results? It seems you have no clue what you're doing.

>>12416612
>one coin is heads
>it's information about two coins
Nope, trying to add fake information about the other coin is cheating. Stop being a schizo.

>>12417175
assuming you know something about ONLY one coin IS fake information hallucinated by a schizo.
You would have to consider either case.

>>12416704
>>X doesn't just imply Y
>X -> Y
I'm sorry you don't understand basic English/logic but that doesn't respond to what I said.

>>12416708
>No one "determined" anything.
Then he's lying and his statement contains no information. The probability remains 1/4. Easy question.

Can the insane schizo stop screaming about "muh fake information", you've been doing it all day and it isnt an argument

>>12416779
>who the fuck is he?
The man in the question.

You really should learn how to read English before posting.

>>12417190
>>12417175
it cannot be information about one coin only. "at least one heads" is false for TT, but true for HT therefore the statement depends on the first coin. on the other hand, "at least one heads" is false for TT, but true for TH, therefore the statement depends also on the second coin. "at least one heads" depends on both coins, so it's not an information about one coin only.

>>12417203
>he
see >>12416779