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A test is designed so that the probability of anyone passing is $80 \%$. Find the probability that among $15$ who took the test, at most 13 will pass.

I know that I can find the probability that exactly $13$ students passed the exam, and the probability that exactly $12$ students passed the exam and so one all the way to $0$ students and then add those probabilities, but I don't know how to find the probability that an exact number of students passed the exam?

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First: It is easier to calculate the probability of 15 or 14 students passing the exam, and to subtract that from 1.

Second, the probability that exactly $n$ students out of 15 pass the exam is:

$P={15\choose n}*0.8^n*0.2^{15-n}$

This is because there are $15\choose n$ ways in which you can have $n$ students all of who pass the exam, leaving $15-n$ students to fail the exam.

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