Question

There are two persons A and B such that the chances of B speaking truth of A and A speaks truth in more than 25% cases. Statement-1: If A and B contradict each other in narrating the same statement with probability 1/2, then it is certain that B never tells a lie. Statement-2: The probability that A speaks truth is 1/2. A. Statement-1 is True, Statement-2 is True, Statement-2 is a correct explanation for Statement-1. B. Statement-1 is True, Statement-2 is True, Statement-2 is not a correct explanation for Statement-1. C. Statement-1 is True, Statement-2 is False. D. Statement-1 is False, Statement-2 is True. Select the correct answer from above options

Answer 1

Correct Answer - A Consider the following events. E =A speaks truth m, F= B speaks truth P ( ( E ∩ ¯¯¯ F ) ∪ ( ¯¯¯ E ∪ F ) ) = 1 2 P ⇒ P ( E ∩ ¯¯¯ F ) + ( ¯¯¯ E ∩ F ) = 1 2 ⇒ ⇒ P € P ( ¯¯¯ F ) + P ( ¯¯¯ E ) + ( F ) = 1 2 ⇒ p × ( 1 − 2 p ) + ( 1 − p ) × 2 p = 1 2 p ⇒ p − 2 p 2 + 2 p − 2 p 2 = 1 2 ⇒ ⇒ 8 p 2 − 6 p + 1 = 0 ⇒ ⇒ ( 4 p − 1 ) ( 2 p − 1 ) = 0 ⇒ p = 1 2 ⇒ P ( E ) = 1 2 P So, statement -2 is ture. Now, P ( E ) = 1 2 ⇒ P ( F ) = 2 p = 1 ⇒ P is a certain event So, B never tells a lie. Hence, statement -1 is also true and statement -2 is a correct explanation for statement -1