you foli a fair coin until you observe at least one tail and

you foli a fair coin until you observe at least one tail and one head, but will flip no more than 4 times. let x be the random variable that is the number of flips you make. determine the probability function for x and determine E(x)

Solution

Assume that p0p0 and p1p1.

If we get a tail on the first toss (probability 1p), the expected number of heads is 1. That is because in this case, the game stops as soon as we get a head.

If we get a head on the first toss, then we already have one head. The expected number of additional tosses until we get a tail is 1p. That is by the standard formula for the mean of a geometrically distributed random variable. All but one of these are additional tosses are heads.

So in that case the expected number of heads is 1+1/(1p)-1=1-p Thus the required expectation is

(1p)(1)+p/(1-p)