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This is an extension of this question: What are Belgium's chances on getting a medal at the 400m finals, given that they have 2 of the 8 athletes?

The original question was: What are the chances of Belgium winning a medal given that they have 2/8 competitors and the probability for a competitor to win is the same for everyone.

The best answer was:

$$ \frac{\binom82-\binom52}{\binom82} = \frac9{14} $$

My question is, what is the probability to win at least 2 medals given that we have 3/8 competitors?

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up vote 1 down vote accepted

I'd go with hypergeometric probability on this one. The probability of exactly two medals is $$ \frac{\binom{3}{2}\binom{5}{1}}{\binom{8}{3}} = \frac{3\cdot 5}{56} $$ because the number of ways for Jamaica to win exactly two medals is the number of ways to pick two Jamaican medalists times the number of ways to pick one non-Jamaican.

The probability of all three of them winning is one in $\binom{8}{3} = 56$, so the answer would be the sum of the two probabilities: $$ \frac{16}{56} = \frac{2}{7} $$

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