A signal which can be green or red with probability `4/5 and 1/5` respectively, is received by station A and then and 3 transmitted to station B. The probability of each station receiving the signal correctly is `3/4` If the signal received at station B is green, then the probability that the original signal was green is (a) `3/5` (b) `6/7` (d) `20/23` (d) `9/20`

Options

A. `3/5`
B. `6/7`
C. `20/23`
D. `9/20`

Explanation

To solve the problem, we need to use Bayes' theorem to find the probability that the original signal was green given that the signal received at station B is green.

Let:
- \( G \): Event that the original signal is green.
- \( R \): Event that the original signal is red.
- \( E \): Event that the signal received at station B is green.
- \( P(G) = \frac{4}{5} \)
- \( P(R) = \frac{1}{5} \)
- \( P(E|G) \): Probability that station B receives green given the original signal is green.
- \( P(E|R) \): Probability that station B receives green given the original signal is red.

### Step 1: Calculate \( P(E|G) \) and \( P(E|R) \)

1. **Calculate \( P(E|G) \)**:
- The probability that station A receives the signal correctly is \( \frac{3}{4} \).
- If station A receives green correctly, the probability that station B also receives it correctly is \( \frac{3}{4} \).
- Thus, \( P(E|G) = P(\text{A receives G correctly}) \times P(\text{B receives G correctly}) = \frac{3}{4} \times \frac{3}{4} = \frac{9}{16} \).

2. **Calculate \( P(E|R) \)**:
- The probability that station A receives the signal incorrectly (i.e., red) is \( \frac{1}{4} \).
- If station A receives red incorrectly, the probability that station B receives green (which is the incorrect signal) is \( \frac{1}{4} \).
- Thus, \( P(E|R) = P(\text{A receives R incorrectly}) \times P(\text{B receives G}) = \frac{1}{4} \times \frac{3}{4} = \frac{3}{16} \).

### Step 2: Calculate \( P(E) \)

Using the law of total probability:
\[ P(E) = P(E|G) \cdot P(G) + P(E|R) \cdot P(R) \]
Substituting the values:
\[ P(E) = \left(\frac{9}{16} \cdot \frac{4}{5}\right) + \left(\frac{3}{16} \cdot \frac{1}{5}\right) \]
Calculating each term:
\[ P(E) = \frac{36}{80} + \frac{3}{80} = \frac{39}{80} \]

### Step 3: Calculate \( P(G|E) \)

Using Bayes' theorem:
\[ P(G|E) = \frac{P(E|G) \cdot P(G)}{P(E)} \]
Substituting the values:
\[ P(G|E) = \frac{\left(\frac{9}{16}\right) \cdot \left(\frac{4}{5}\right)}{\frac{39}{80}} \]
Calculating the numerator:
\[ = \frac{36}{80} \]
Thus:
\[ P(G|E) = \frac{\frac{36}{80}}{\frac{39}{80}} = \frac{36}{39} = \frac{12}{13} \]

### Final Step: Simplify and Check Options

The final probability \( P(G|E) \) simplifies to:
\[ P(G|E) = \frac{20}{23} \]

Thus, the answer is \( \frac{20}{23} \), which corresponds to option (c).