Grinstead and Snell’s Introduction to Probability, Chapter 11, #3

Consider Example 11.4 with a = .5 and b = .75. Assume that the President says that he or she will run. Find the expected length of time before the first time the answer is passed on incorrectly.

P(yes, no)=a P(yes, yes)=1-a

\[P(x=n)=(1-a)^{n-1} a\] \[ E(x)=\Sigma_{n=1}^{\infty} n(1-a)^{n-1} a\]

\[ =a \Sigma_{n=1}^{\infty} n(1-a)^{n-1} \] \[ \frac{a}{(1-(1-a))^{2}}= \frac {a}{a^{2}}=\frac{1}{a} =\frac{1}{0.5}=2 \]