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I'm trying to determine the probability that I'll "place" within certain percentage in a given contest against other competitors. The two specific examples I can provide that I do not know how to determine the answer to are as follows:

Example 1: I'm entering a "head to head" contest. I know that I place 1st (win) head to head contests 70% of the time. The individual I'm competing against places 1st (wins) head to head contests 60% of the time. How do I determine the probability that I'll beat this competitor in a head to head contest?

Example 2: I'm entering a contest where there are 10 competitors. To win in this contest, I need to place in either 1st, 2nd, 3rd or 4th place. I know that I typically place within the top 37% of all contests of this type (average figure for all contests of this type I've entered). I know that the other 9 competitors entering have the following overall placement %'s: top 10%; top 20%; top 30%; top 40%; top 50%; top 60%; top 70%; top 80%; top 90%. How do I determine the probability that I'll place in the top 40% (in either 1st through 4th place).

I know it is typically good etiquette to explain how far I've gotten in a solution, but I honestly do not even know where to begin. Appreciate any assistance this community can offer.

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  • $\begingroup$ Should mention that there is no possibility for any one player to "team up" with another player to make it more difficult for yet another player to win. Each person in each example above is playing by themselves against the others with no ability to influence the other contestants outcome. $\endgroup$ – Kevin Stevens Sep 22 '16 at 18:00
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I can answer the first question. Here we have two players, PI and PII. Suppose there is one more, "average" player A. PI wins against A 7 times out of 10. PII wins against A 6 times out of 10.

We can say that PI is 7/3 times as good as A, and PII is 6/4 times as good as A. Then it's trivial to answer the question by how much PI is better than PII. That quantity will also be equal to the odds that PI beats PII:

Odds = (7/3) / (6/4) = 14/9, meaning that the probability of PI beating PII is 14 / (14 + 9) = 0.6086.

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