Communication, Networking, Signal and Image Processing (CS)

Question 1: Probability and Random Processes

January 2002

1. (20 pts)

Given two coins; the first coin is fair and the second coin has two heads. One coin is picked at random and tossed two times. It shows heads both times. What is the probability that the coin picked is fair?

## Solution 1

• $ F=\left\{ \text{fair coin is selected}\right\} $

• $ S=\left\{ \text{the coin that has two heads is selected}\right\} $

• $ H2=\left\{ \text{heads are shown both times}\right\} $

• $ P\left(H2|F\right)=\frac{1}{4},\; P\left(H2|S\right)=1,\; P\left(F\right)=P\left(S\right)=\frac{1}{2}. $

• By using Bayes' theorem,$ P\left(F|H2\right)=\frac{P\left(H2|F\right)P\left(F\right)}{P\left(H2|F\right)P\left(F\right)+P\left(H2|S\right)P\left(S\right)}=\frac{P\left(H2|F\right)}{P\left(H2|F\right)+P\left(H2|S\right)}=\frac{\frac{1}{4}}{\frac{1}{4}+1}=\frac{1}{5}. $