In a quiz with 13 people, what are the probabilities that exactly 12, exactly 11 and exactly 10 people answer correctly?

Question

A group of 13 people, p1, p2, p3, ... , p13 answers the same question. The probability that a person answers correctly are known but differs between the people. For example the probability that p1 answers correctly is 0.68, for p2 it's 0.23 and so on. How can I calculate the probabilities that, for example,  exactly 10 people get the question correct?

Sameer Baheti · Answer

If the answers are independent events,
$$p=prod_{i=k}^{l}p_iprod_{jneq i}(1-p_j)$$
is the probability that only persons $k,k+1,ldots,l$ answer correctly, where $p_i$'s are the probability of correctly answering.
Write the same for all ${13choose l-k+1}$ combinations and add all of them to get the final probability.

In a quiz with 13 people, what are the probabilities that exactly 12, exactly 11 and exactly 10 people answer correctly?

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