1989 AIME Problems/Problem 5
When a certain biased coin is flipped five times, the probability of getting heads exactly once is not equal to and is the same as that of getting heads exactly twice. Let , in lowest terms, be the probability that the coin comes up heads in exactly out of flips. Find .
Denote the probability of getting a heads in one flip of the biased coin as . Based upon the problem, note that . After canceling out terms, we get , so . The answer we are looking for is , so .
Denote the probability of getting a heads in one flip of the biased coins as and the probability of getting a tails as . Based upon the problem, note that . After cancelling out terms, we end up with . To find the probability getting 3 heads, we need to find (recall that cannot be 0). The result after simplifying is , so .
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