Question

The input X to the Binary Symmetric Channel (BSC) shown in the figure '1' with probability 0.8.
The crossover probability is 1/7. If the received bit Y = 0,. the conditional probability that '1' was transmitted is .

• A. 0.4

Answer 1

The correct option is A 0.4

Let y = {Received bit is y = 0}
$E_1=\left\{\text{Input x = 0}\right\},E_2=\left\{\text{Input X = 1}\right\}$

$P\left(E_1\right)=P(X=0)=0.2,$
& $P\left(E_2\right)=P(X=1)=0.8$

$P\left(Y/E_2\right)=P\left[\frac{Y=0}{X=1}\right]=\frac{1}{7}$

$P\left(Y/E_1\right)=P\left[\frac{Y=0}{X=0}\right]=\frac{6}{7}$

So by Baye's theorem

$P\left[\frac{X=1}{Y=0}\right]=P\left(\frac{E_2}{Y}\right)$

$=\frac{P\left(E_2\right).P\left(Y/E_2\right)}{P\left(E_1\right).P\left(Y/E_1\right)+P\left(E_2\right).P\left(Y/E_2\right)}$

$=\frac{0.8\times \frac{1}{7}}{0.2\times \frac{6}{7}+0.8\times \frac{1}{7}}=0.4$