A fair coin is tossed repeatedly until two consecutive heads appear. Let $P(n)$ be the probability that the process stops at exactly the $n$-th toss. Find the value of ${\sum }_{n=2}^{\infty }P(n)$.

Solution

1. Let $E$ be the event that the sequence of tosses eventually contains two consecutive heads.
2. The probability of not getting two consecutive heads in $n$ tosses approaches $0$ as $n\to \infty$.
3. Since the process must terminate with probability $1$, the sum of probabilities of all disjoint events $P(n)$ is $1$.
4. Hence the answer is (B).