Question

A biased coin has probability p to land heads and q = 1 − p to land tails. The coin is flipped until the first occurrence that differs from the initial flip. What is the number of flips required, on average?

Answer 1

The coin is flipped until the first occurrence differs from the initial flip. Therefore the average number of flips here is obtained as:

HT, TH, HHT, TTH, HHHT, TTTH, .... and so on..

The expected number of flips required here is computed as:

Multiplying both sides by 0.5, we get here:

Subtracting the last equation from the second last one, we get here:

Therefore 3 is the required expected number of flips here.