A gives a correct answer in 75% of cases and B in 80% of cases, independently of each other. A question whose answer is either “yes” or “no” is put to both of them, and they give the same answer. The probability that their common answer is correct is
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \(\displaystyle\frac{12}{13}\)
The answer is “yes” or “no”, so A and B give the same answer when both are correct or when both are wrong. As they answer independently:
\( \displaystyle P(\text{both correct}) = \frac{3}{4}\times\frac{4}{5} = \frac{12}{20} \)
\( \displaystyle P(\text{both wrong}) = \frac{1}{4}\times\frac{1}{5} = \frac{1}{20} \)
\( \displaystyle P(\text{same answer}) = \frac{12}{20}+\frac{1}{20} = \frac{13}{20} \)
By conditional probability:
\( \displaystyle P(\text{correct} \mid \text{same answer}) = \frac{\frac{12}{20}}{\frac{13}{20}} = \frac{12}{13} \)
Why the other options are incorrect
Option (1) \(\displaystyle\frac{3}{5}\) — Incorrect. This is the chance that both answer correctly, without using the information that their answers are the same.
Option (2) \(\displaystyle\frac{13}{20}\) — Incorrect. This is the chance that they give the same answer.
Option (3) \(\displaystyle\frac{1}{13}\) — Incorrect. This is the chance that their common answer is wrong.