A signal which can be green or red with probability \frac{4}{5} \textrm{ }\textrm{ } and \textrm{ }\textrm{ } \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

Engineering

Mathematics

Total Probability Theorem and Bayes Theorem

Question

A signal which can be green or red with probability $\frac{4}{5} and \frac{1}{5}$ 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

$\frac{6}{7}$

$\frac{3}{5}$

$\frac{9}{20}$

$\frac{20}{23}$

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Probability of B receiving green signal

$P (G) = \frac{4}{5} [\frac{3}{4} \times \frac{3}{4} + \frac{1}{4} \times \frac{1}{4}] + \frac{1}{5} [\frac{3}{4} \times \frac{1}{4} + \frac{1}{4} + \frac{3}{4}] = \frac{1}{2} + \frac{3}{8} = \frac{3}{40}$

Probability that signal related was green while B received Green $= \frac{\frac{1}{2}}{\frac{1}{2} + \frac{3}{40}} = \frac{20}{23}$ Ans.

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