Communication, Networking, Signal and Image Processing (CS)
Question 1: Probability and Random Processes
August 2001
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?
(b) What is the probability that this experiment terminates with an even number of coin tosses?
Solution 1
(a)
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)
$ 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}. $
