1. (10 Points)
Consider the following random experiment: A fair coin is repeatedly tossed until the same outcome (H or T) appears twice in a row.
(a)
What is the probability that this experiment terminates on or before the seventh coin toss?
Let N be the number of toss until the same outcome appears twice in a row.
| $ N $th | $ \left(N - 1\right) $th | $ \left(N - 2\right) $th | $ \left(N - 3\right) $th | $ \cdots $ |
|---|---|---|---|---|
| H | H | T | H | $ \cdots $ |
| T | T | H | T | $ \cdots $ |
$ P\left(\left\{ N=n\right\} \right)=\frac{2}{2^{n}}=\frac{1}{2^{n-1}}\text{ for }n\geq2. $
$ P\left(\left\{ N\leq7\right\} \right)=\sum_{k=2}^{7}\frac{1}{2^{k-1}}=\sum_{k=1}^{6}\left(\frac{1}{2}\right)^{k}=\frac{\frac{1}{2}\left(1-\left(\frac{1}{2}\right)^{6}\right)}{1-\frac{1}{2}}=1-\frac{1}{64}=\frac{63}{64}. $
(b)
What is the probability that this experiment terminates with an even number of coin tosses?
$ P\left(\left\{ N\text{ is even}\right\} \right)=\sum_{k=1}^{\infty}\frac{1}{2^{2k-1}}=2\sum_{k=1}^{\infty}\left(\frac{1}{4}\right)^{k}=2\cdot\frac{\frac{1}{4}}{1-\frac{1}{4}}=2\cdot\frac{1}{3}=\frac{2}{3}. $
