Question

A source produces three equiprobable symbals $x_1,x_2$ and $x_3$ which are transmitted on a channel shown in the figure below



If the maximum likelyhood decision rule is used, then the minimum error probability of the channel ii equal to

• A. 0.533
• B. 0.919
• C. 0.6911
• D. 0.4125

Answer 1

The correct option is A 0.533

$P_o=1-P_e$
Now, since we are using ML criterion, and the input symbols are equally likely, then we can directly choose the output based on the maximum value of the tranmission probabilites.
$P_o=P\left(x_1\right)P\left(y_1|x_1\right)+P\left(x_2\right).P\left(y_2|x_2\right)+P\left(x_3\right).P\left(y_2|x_3\right)$
$=\frac{1}{3}[0.5+0.4+0.5]=\frac{7}{15}$
$\therefore P_e=1-\frac{7}{15}=\frac{8}{15}=0.533$