In: Math
A fair coin is tossed repeatedly until it has landed Heads at least once and has landed Tails at least once. Find the expected number of tosses.
Let be the probability of landing Heads.
The experiments will terminate in tosses,
1) When the toss is a Head preceded by Tails. Whose probability is
2) When the toss is a Tail preceded by Heads. Whose probability is
The above 2 events are disjoint.
The probability that the experiments will terminate in tosses is
Here . Hence,
The expected tosses is
Note that is the expected value of a Geometric distribution which is 2. Thus,