football probabilities

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 0–B ties with probability .1; B wins with probability .88; A wins with probability .02.

if A scored 3–B ties with probability .3; B wins with probability .6; A wins with probability .1.

if A scored 6–B ties with probability .01; B wins with probability .4; A wins with probability .59.

if A scored 7–B ties with probability .3; B wins with probability .1; A wins with probability .6.

if A scored 8–B 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; they’re tied after the first round.

b) Find the probability that A wins.

c) Find the expected number of rounds

I have calculated a to be .345, .436 and .219, but I'm not sure what to do for b or c.

Thanks!

• I would really appreciate any help. – No Name Oct 7 '13 at 16:53

b. Probability that A wins.

Ultimately A or B has to win. Therefore you do not need to consider the tie result so:

$$P(A)=\frac{P(A_1)}{P(A_1)+P(B_1)}$$

c. Expected number of rounds.

This is a geometric distribution. For the expected number of rounds (i.e. including the round with a winner) you use the first form. So, the expected value is another way of saying the mean of the distribution:

$$E=\frac{1}{p}$$

where $p$ is the probability of a result in a given round i.e. $p=P(A_1)+P(B_1)$

what you need to do is look at the ways in which a can win. A has an arbitrary number of points given so you look at the outcomes for B and there it says the probability of A winning given that outcome. So the conditional probability of A winning given A having that number of points is p(a scores x points)*p(A wins given a scores x points) Then just add them up.

I'm not an expert in american football so I assume if they tie then they go to another round. Then you need to just keep doing similar stuff I think. Though this is more arduous/

• I'm pretty sure it would be a geometric series because the probability that A wins in two round would be P(tie in first) *P(A wins in 2nd)..and so on for infinity. But how would write this as a geometric series? – No Name Oct 8 '13 at 2:01