Question

Compute the expected value of the random number of coin tosses until a run of k successive heads occurs when the tosses are independent and each lands on heads with a probability 1/2.

A. 2^(k−1)

B. 2^(k) + 2

C. 2^(k+1) − 2

D. 2^(k+1)

E. Does not exist

Answer 1

Let e be the expected number of tosses.now on

Start tossing. If we get a tail immediately (probability 1/2) then the expected number of getting consecutive k heads will be e+1. If we get a head then a tail (probability 1/4), then the expected number is e+2. Continue …. If we get 4 heads then a tail, the expected number is e+5. Continue the process we get