Question

A signal which can be green or red with probability $\frac{2}{3}$ and $\frac{1}{5}$ respectively, is received by station A and then transmitted to station B. The probability of each station receiving the signal correctly is $\frac{3}{4}$.If the signal received at station B is green, then the probability that the original signal was green is

• A.
• B.
• C.
• D.

Answer 1

The correct option is C

Even G = original signal is green
$E_1$ = A receives the signal correct
$E_2$ = B receives the signal correct
E = Signal received by B is green
P = (Signal received by B is green)
$=P\left(G{E}_1{E}_2\right)+P\left(G\overline{E_1}\overline{E_2}\right)+P\left(\overline{G}{E}_1\overline{E_2}\right)+P\left(\overline{G{E}_1}{E}_2\right)P\left(E\right)=\frac{46}{5\times 16}P\left(\frac{G}{E}\right)=\frac{\frac{40}{5}\times 16}{\frac{46}{5}\times 16}=\frac{20}{23}$