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

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 green is
  1. $\frac{3}{5}$
  2. $\frac{6}{7}$
  3. $\frac{20}{23}$
  4. $\frac{9}{20}$

Solution

From the tree-diagram it follows that
$ \begin{gathered} P\left(B_G \mid G\right)=\frac{10}{16}=\frac{5}{8} \\ \therefore \quad P\left(B_G \cap G\right)=\frac{5}{8} \times \frac{4}{5}=\frac{1}{2} \\ P\left(G \mid B_G\right)=\frac{\frac{1}{2}}{P\left(B_G\right)}=\frac{1}{2} \times \frac{80}{46}=\frac{20}{23} \end{gathered} $

Asked in: JEE Advanced 2010 (Paper 2)

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