(a) Andrei flips a coin over and over again until he gets a tail followed by a head, then he quits. What is the expected number of coin flips?
(b) Bela flips a fair coin over and over again until she gets two tails in a row, then she quits. What is the expected number of coin flips?
Hello,
I have been having some trouble with this problem. I have tried to use state diagrams, but bonked my head on the table, because obviously, that wouldn't work. I have not yet been able to find any method in doing this. Any help is appreciated
Best Answer
In the first case (a) the sequence of the outcomes is something like $H^j T^k H$ with $j\geq 0$ and $k\geq 1$. Such a sequence has length $j+k+1$, hence the expected number of coin flips is given by: $$\sum_{j\geq 0}\sum_{k\geq 1}\frac{j+k+1}{2^{j+k+1}}=\sum_{h,k\geq 1}\frac{h+k}{2^{h+k}}=4.$$ In the second case (b), the sequence of outcomes is a string over $\{H,TH\}$ plus a $TT$ suffix. The number of strings of length $N$ over $\Sigma=\{H,TH\}$ is given by the $(N+1)$-th Fibonacci number $F_{N+1}$, hence the expected number of coin flips is given by: $$ 2+\sum_{N=0}^{+\infty}\frac{N\cdot F_{N+1}}{2^{N+2}}=6.$$