Question

An unbiased coin is tossed until a head appears and then tossed until a tail appears. If the tosses are independent, what is the probability that a total of exactly n tosses will be required?

Answer 1

Let X be the number of tosses required to get the first heads and Y be the number of tosses required to get the first tail after first heads be Y. Then the probability that a total of n tosses would be required is computed here as:

= P(X = 1)P(Y = n - 1) + P(X = 2)P(Y = n - 2) + .... + P(X = n - 1)P(Y = 1)

The number of terms here are n - 1. Therefore the probability here is computed as:

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