# Thread: conditional probability

1. ## conditional probability

a football match is played between Team 1 and Team 2. The probability that team 1 wins is $p$ and the for team 2 to win the probability is $q$; finally to draw is $1 - p - q$. In the event of a draw the two teams must play a rematch with the same probabilities as before. In the event of a second draw there is a penelty shoot out whith each team having a 50% chance of winning.

In terms of $p$ and $q$ what is the probability that Team 1 defeats Team 2?

Letting T1, T2 and D be the events of Team 1 winning, Team 2 winning and a draw respectively. I draw a tree diagram and deduce that $P(T1) \cup P(T1\mid D) \cup P(T1 \mid D)$ which is equal therefore to
p + p(1 - p - q) + 0.5(1 - p - q)(1 - p - q). There are further questions but I don't feel confident that my answer is correct....

Thanks for any help in advance.
D

2. Hello, dojo!

$\text{A football match is played between Team 1 and Team 2.}$
$\text{The probability that team 1 wins is }p\text{, and that team 2 wins is }q.$
$\text{The probability that they draw is }1 - p - q$.

$\text{If they draw, they rematch with the same probabilities as before.}$
$\text{In the event of a second draw, there is a sudden-death shootout}$
$\text{with each team having a 50\% chance of winning.}$

$\text{In terms of }p\text{ and }q\text{, what is the probability that Team 1 wins?}$

There are three ways that Team 1 can win.

[1] Team 1 wins the match.
. . . $P(\text{Team 1 wins}) \:=\:p$

[2] They draw the match and Team 1 wins the rematch.
. . . $P(\text{Team 1 wins}) \:=\:(1-p-q)p$

[3] They draw the match, draw the rematch, and Team 1 wins the shootout.
. . . $P(\text{Team 1 wins}) \:=\:(1-p-q)^2(\frac{1}{2})$

Therefore:

. . $P(\text{Team 1 wins}) \;=\;p + p(1-p-q) + \frac{1}{2}(1-p-q)^2$

. . . . . . . . . . . . . . $=\;p + p - p^2 - pq + \frac{1}{2}(1 - 2p - 2q + p^2 + 2pq + q^2)$

. . . . . . . . . . . . . . $=\;p + p - p^2 - pq + \frac{1}{2} - p - q + \frac{1}{2}p^2 + pq + \frac{1}{2}q^2$

. . . . . . . . . . . . . . $=\; p - q - \frac{1}{2}p^2 + \frac{1}{2}q^2 + \frac{1}{2}$