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 given, then the probability that the original signal is green is:

• A. \( \frac{3}{5} \)
• B. \( \frac{6}{7} \)
• C. \( \frac{20}{23} \)
• D. \( \frac{9}{20} \)

Hint:

Bayes' Theorem helps in finding the probability of an event based on conditional probabilities.

Answer 1

The Correct Option is C

Step 1: Use Bayes' Theorem to find the probability that the signal is green, given the signal received at station B.
Step 2: The probability of receiving a green signal correctly at station B is \( P(\text{green received}) = \frac{4}{5} \times \frac{3}{4} \), and the probability of receiving a red signal correctly is \( P(\text{red received}) = \frac{1}{5} \times \frac{3}{4} \).
Step 3: Use Bayes' formula to find the conditional probability, which gives \( \frac{20}{23} \).

Final Answer: \[ \boxed{\frac{20}{23}} \]