Probability
Bayes' Theorem
IIT-JEE 2010
Grade 12
Question:
A signal which can be green or red with probabilities $\frac{4}{5}$ 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, the probability that the original signal was green is:
(1) $\frac{20}{23}$
(2) $\frac{23}{40}$
(3) $\frac{3}{8}$
(4) $\frac{5}{8}$
Step-by-Step Solution
Key Concept: This is a Bayes' theorem problem — first compute $P(B \text{ receives green} \mid \text{original green})$ and $P(B \text{ receives green} \mid \text{original red})$ by considering both stations relaying correctly or both incorrectly.
The detailed step-by-step mathematical proof is available inside the Mathbee app workspace.
Correct Answer: (1)