Question

A telegraphic communication system transmits the signals dash and dot. Assume that $\frac{1}{3}$ of the dashes are changed to dots and $\frac{2}{5}$ of the dots are changed to dashes. Also suppose that the ratio of the transmitted dashes and dots is $3:5$.
The probability that the transmitted signal is a dot if the signal received is a dot, is

• A. $\frac{6}{8}$
• B. $\frac{5}{8}$
• C. $\frac{4}{8}$
• D. $\frac{3}{8}$

Answer 1

Answer: (A)

A : event that a dash is received
B : event that dot is received
$H_1$: transmitted signal was a dash
$H_2$ : transmitted signal was a dot.
$P\left(B/H_1\right)=\frac{1}{3};P\left(A/H_2\right)=\frac{2}{5}$
$P\left(A/H_1\right)=\frac{2}{3};P\left(B/H_2\right)=\frac{3}{5}$
Signals transmitted dash $=3k$ (say) and signals transmitted dot $=5k$ (say)
$\Rightarrow P\left(H_1\right)=\frac{3}{8};P\left(H_2\right)=\frac{5}{8}$
$P(A)=P\left(A\cap H_1\right)+P\left(A\cap H_2\right)$
$=P\left(H_1\right)\cdot P\left(A/H_1\right)+P\left(H_2\right)\cdot P\left(A/H_2\right)$

Now $P(A)=\frac{3}{8}\cdot \frac{2}{3}+\frac{5}{8}\cdot \frac{2}{5}=\frac{1}{2}$
$P(B)=P\left(B\cap H_1\right)+P\left(B\cap H_2\right)$
$=P\left(H_1\right)\cdot P\left(B/H_1\right)+P\left(H_2\right)\cdot P\left(B/H_2\right)$
$P(B)=\frac{3}{8}\cdot \frac{1}{3}+\frac{5}{8}\cdot \frac{3}{5}=\frac{1}{2}$
Now
$P\left(H_2/B\right)=\frac{P\left(H_2\cap B\right)}{P(B)}=\frac{P\left(H_2\right)\cdot P\left(B/H_2\right)}{P(B)}=\frac{\frac{5}{8}\cdot \frac{3}{5}}{\frac{1}{2}}=\frac{3}{8}\cdot \frac{2}{1}=\frac{3}{4}=\frac{6}{8}$