Suppose that we flip a fair coin until either it comes up ta

Suppose that we flip a fair coin until either it comes up tails twice or we have flipped it six times. What is the expected number of times we roll the die?

Solution

Let X be random variable \'number of times \' the roll will die

Here X takes values 2,3,4,5 and 6

when X=2; the first two flips are tails

X=3 ;one tail in first two flips and last flip is tail ; X=4 there is one tail in the first three flips and last flip is tail

X=5 ; one tail in the first four flips and fifth one is tails ; X=6 ; first five flips only one tail or no tail and then experiment stopped

probability of a head =1/2 ; probability of a tail = 1/2

Expected number of times is 3.75