Can this conditional probability be answered using Bayesian Theorem (or at all) with the information given

Question

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.

Answer 1

It is not as simple as saying that $A_1 = C_1 * D_1$ (different teams may have different probabilities of winning given that they won the first game.)

For this problem your space consists of four events.

1. AA = Team A scores first and wins.
2. AB = Team A scores first and loses.
3. BA = Team B scores first and loses.
4. BB = Team B scores first and wins.

Write your known information in terms of those events, and you should have enough information to find the probabilities of each of the events. (You should probably also go back and verify that your Bayesian argument does not work once you have all the probabilities.)

Answer 2

One possibility to organize your information is to put it in a two-by-two table where rows are A wins, B wins; and columns are A scores first, B scores first. (You could also put your info in a 2-circle Venn diagram).

Then see where it leads.

Answer 3

Thanks very much - both answers were very helpful.

Once I constructed a 2x2 table of the possibilities it became more apparent that I had all the information I needed. I could write all the known probabilities down as the result of a set of simultaneous equations and then solve these to get my unknown probabilities.

Using the notation I had in my original question, I arrived at the following answers:

A1 (prob Team A scores first and wins) = (B1 - C2 + D1)/2

and similarly,

A2 (prob Team B scores first and wins) = (B2 - C1 + D1)/2

Thanks for your help!