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?
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.
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