I try to solve the following problem, and for some reason my answer does not agree with the one from the solutions.
Consider the Bernoulli process, which we stop after first occurence of consecutively success and a failure. Let $X$ be a random variable describing the number of trials in this experiment. The aim is to compute $EX$.
So I am using conditional expected value. I have the following events:
- $A$: failure in the first trial;
- $B$: Success in the first trial, failure in the second;
- $C$: Success in both first and second trials.
This gives me an equation: $$ EX=E(X|A)P(A)+E(X|B)P(B)+E(X|C)P(C) $$ giving $$ EX=(1+EX)(1-p)+2p(1-p)+(1+EX)p^2 $$ and solving this for $EX$ gives an answer $EX=1+\frac{1}{p(1-p)}$. However, the textbook says that the answer is $\frac{1}{p(1-p)}$. I suppose that the mistake is in assuming that $E(X|C)=1+EX$, but I cannot think about other method. Any help will be appreciated!
Let the answer be $E$. Consider the first trial. If it is $S$ then we are just waiting for the first $F$. We expect that to take $\frac 1{1-p}$ trials (from that point). If the first toss is $F$, then the game resets (so you again expect $E$ trials from this point).
Thus: $$E=p\times \left(1+\frac 1{1-p}\right)+(1-p)\times (E+1)\implies E=\frac 1{p(1-p)}$$