I have a conditional probability problem I'm unsure can be answered given the information I have - as such I'm unsure if Bayesian Theorem is the way to answer it, or if the answer is staring at me in the face and I can't spot it.
It's easiest to think of this problem in terms of a game of baseball, with two teams - A and B. I want to work out:
Pr(A scores first and A wins) (A1) - and similarly
Pr(B scores first and B wins) (A2)
Obviously these two scenarios alone won't sum to 100% as there are the possibilities that a team could score first and go on to lose. Also worth keeping in mind that in a baseball setting Team A will bat first and have first chance to score.
I have the following known information:
Pr(A Wins) = 0.58 (B1)
Pr(B Wins) = 0.42 (B2)
Pr(A Scores First) = 0.51 (C1)
Pr(B Scores First) = 0.49 (C2)
Pr(The team who scores first wins) = 0.75 (D1)
Pr(The team who scores first loses) = 0.25 (D2)
Is it as simple as saying that A1 = C1 * D1 (ie, the prob of team A scoring first by the prob that the team who scores first wins is the prob that team A will score first and win).
In the back of my mind it seems to me this isn't valid as there is some dependency here, hence I'm unsure whether I need to use Bayes Theorem - or is it the case there isn't enough information here?
Thanks in advance.