/sci/ - Science & Math

I strongly believe in 50% because given X, Y (X=1 ^ Y=2 or X=2 ^ Y=1) say hit X is given to be the critical one, no matter what Y still has an independently 50% chance of a critical. Hence 50% overall.
However, my professor told me to just apply conditional probability's definition from which I got the following.

> A = both crit
> B = at least one crit
> P (A ^ B) = P (B) x P (A | B)
> P (A | B) = P (A ^ B) / P (B)
> P (A | B) = 0.25/0.75 = 1/3

What is the fallacy I made in my argument in the first line arguing for 50%, 1/3fags?
Or rather did I make a mistake in the mathematical solution for 1/3? Did I get P (B) wrong? 50%ers?

Because, if I redo the calculation with A and B as the following (more similar to my argument above for 50%):

> A = first hit crit
> B = second hit crit
> P (A ^ B) = P (B) x P (A | B)
> P (A | B) = P (A ^ B) / P (B)
> P (A | B) = 0.25/0.5 = 0.5
Which works because we have two cases: A is given to be crit, or B is given to be crit. These cases are symmetrical. So in any case the chances are 50%.

What is the fallacy in my second mathematical argument, 1/3'ers?

Please. I'm dying to know.