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I have the following problem:

Suppose that the probability of a football team (T) to win the match with 2-1 is p1 and, given that T scores a goal, the probability that the best player of the team scored it is p2. What is the probability that the team T wins with 2-1 and the best player scored the 1st goal?

My thinking was to use the conditional probability formula to get the intersection.

Let

A = event of T willing the football match with 2-1- P(A) = p1

B = event of the best player scoring in the match

I know P(B|A) - probability of the best player scoring in the match given that T won the football match with 2-1 is p2.

Then P(A ∩ B) = probability of T winning the match by 2-1 and the best player scoring the goal and then use the conditional probability formula:

P(B | A) = P(B ∩ A)/P(A)

p2 = P(B ∩ A)/p1

P(A ∩ B) = p1 × p2.

But in the question in says to calculate the probability that T wins the match by 2-1 and the best player scores the 1st goal. How can I find the probability of the best player score the first goal? Should I assume it follows a distribution?

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    $\begingroup$ Please add the self-study tag & read its wiki. Then tell us what you understand thus far, what you've tried & where you're stuck. We'll provide hints to help you get unstuck. Please make these changes as just posting your homework & hoping someone will do it for you is grounds for closing. $\endgroup$ May 14 at 15:36
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    $\begingroup$ You added the self-study tag. Now it's time for the rest of the steps Kjetil outlined. $\endgroup$ May 15 at 16:53

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How can I find the probability of the best player score the first goal?

This is given in the question

"given that T scores a goal, the probability that the best player of the team scored it is p2"

I think you can assume that the question is about the best player scoring the first goal for the team and not the first goal in the entire match (for the latter case you would indeed need to make assumptions about the probability that it is team T and not their opponent that scores the first goal).

Probably you may also assume that the probability $p_2$ that the best player scores the goal is independent on whether it is the first, second, third , etc. goal of the team.

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  • $\begingroup$ If I want to calculate the probability of the team scores the first goal in the entire match how can I do that? $\endgroup$
    – alan watt
    May 16 at 12:01
  • $\begingroup$ @alanwatt there is no sufficient information for that and you can only calculate this by making additional assumptions, for instance that the goal orders STT, TST and TTS are equally probable. $\endgroup$ May 16 at 12:09
  • $\begingroup$ I accidentally put the team in my previous comment but I was meant the best player. I would be interested to see how to model the best player scoring the first goal. $\endgroup$
    – alan watt
    May 16 at 15:28

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