jmc

Question

When a certain biased coin is flipped five times, the probability of getting heads exactly once is not equal to 0 and is the same as that of getting heads exactly twice. Let ij, in lowest terms, be the probability that the coin comes up heads in exactly 3 out of 5 flips. Find i+j.

Accepted Answer

Denote the probability of getting a heads in one flip of the biased coin as h. Based upon the problem, note that 51(h)^1(1-h)^4 = 52(h)^2(1-h)^3. After canceling out terms, we get 1 - h = 2h, so h = 13. The answer we are looking for is 53(h)^3(1-h)^2 = 10(13)^3(23)^2 = 40243, so i+j=40+243=283.