Probability that A speaks truth is \(\displaystyle\frac{4}{5}\). A coin is tossed. A reports that a head appears. The probability that actually there was head is:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \(\displaystyle\frac{4}{5}\)
Let \(E_1\): head appears, \(E_2\): tail appears, and R: A reports a head. Then
\( \displaystyle P(E_1) = P(E_2) = \frac{1}{2} \)
A reports a head when it is a head and A tells the truth, or when it is a tail and A lies:
\( \displaystyle P(R \mid E_1) = \frac{4}{5},\quad P(R \mid E_2) = \frac{1}{5} \)
By Bayes’ theorem:
\( \displaystyle P(E_1 \mid R) = \frac{P(E_1)P(R \mid E_1)}{P(E_1)P(R \mid E_1) + P(E_2)P(R \mid E_2)} \)
\( \displaystyle = \frac{\frac{1}{2}\times\frac{4}{5}}{\frac{1}{2}\times\frac{4}{5} + \frac{1}{2}\times\frac{1}{5}} \)
\( \displaystyle = \frac{4}{4+1} = \frac{4}{5} \)
NCERT (2026-27 edition):
Bayes' Theorem
