First slide

Theorems of probability

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

Moderate

A

$\frac{3}{5}$
B

$\frac{6}{7}$
C

$\frac{20}{23}$
D

$\frac{9}{20}$

Solution

Even G - original signal is green
E₁ = A receives the signal correct
E₂= B receives the signal correct
E - Signal receive d by B is green
P (Signal received by B is green )
$= P (G E_{1} E_{2}) + P (G {\bar{E}}_{1} {\bar{E}}_{2}) + P (\bar{G} E_{1} {\bar{E}}_{2}) + P (\bar{G} {\bar{E}}_{1} E_{2})$
$\begin{array}{l} P (E) = \frac{46}{5 \times 16} \\ P (G / E) = \frac{40 / 5 \times 16}{46 / 5 \times 16} = \frac{20}{23} \end{array}$

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