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Thread: conditional probability

  1. #1
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    conditional probability

    a football match is played between Team 1 and Team 2. The probability that team 1 wins is $\displaystyle p$ and the for team 2 to win the probability is $\displaystyle q$; finally to draw is $\displaystyle 1 - p - q$. 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 $\displaystyle p$ and $\displaystyle q$ 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 $\displaystyle 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
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  2. #2
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    Hello, dojo!

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

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

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

    There are three ways that Team 1 can win.


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


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


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


    Therefore:

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

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

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

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

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