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October 5th, 2013, 11:10 AM   #1
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Probability question-- need help

A and B are college football teams that have gone into overtime.

In the first round A will go first with the following possible outcomes: no score; 3 points; 6 points; 7 points; 8 points; a turnover where B wins (note in this case the game ends immediately). The probabilities of these happening are: .2, .3, .1, .3, .09, .01.

B then follows with the following conditional outcomes:
if A scored 0B ties with probability .1; B wins with probability .88; A wins with probability .02.
if A scored 3B ties with probability .3; B wins with probability .6; A wins with probability .1.
if A scored 6B ties with probability .01; B wins with probability .4; A wins with probability .59.
if A scored 7B ties with probability .3; B wins with probability .1; A wins with probability .6.
if A scored 8B ties with probability .2; A wins with probability .8
If the teams are tied after the first round, they go to a second round and continue until a team wins.

a) Find the probability that: A wins in the first round; B wins in the first round; theyre tied after the first round.
b) Find the probability that A wins.
c) Find the expected number of rounds.

I figured out part a) 0.345, 0.436, and 0.219. Not sure how to do parts b and c. Help! thanks
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October 7th, 2013, 04:08 PM   #2
jks
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Re: Probability question-- need help

Hi MathNoob123, and welcome to the forums.

I agree with your calculations for part a).

Let's use those probabilities to find b) and c):



The probability of A winning is the probability of A winning in the first round (0.345) plus that of the second round, and so on. The probability of a tie (0.219) factors in as follows:





The infinite sum inside the parenthesis is:

so:






The expected value is the sum of the round number times that round's probability of ending the game:





The infinite sum inside the parenthesis is:

so:



One comment, although I am not an expert:
The statement "if A scored 0B ties with probability .1; B wins with probability .88; A wins with probability .02." seems incomplete. The only way that A can win in that round is if B turns the ball over, and that is not specifically stated. I assumed that this is the case.
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