IIT-JEE 2007 · Paper 1 · Mathematics Q54

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MathematicsAssertion and reason+3 / -1Question 54 of 66

Let $H_1,H_2,\ldots,H_n$ be mutually exclusive and exhaustive events with $P(H_i)>0$ , $i=1,2,\ldots,n$ . Let $E$ be any other event with $0<P(E)<1$ .

STATEMENT-1: $P(H_i\mid E)>P(E\mid H_i)\cdot P(H_i)$ for $i=1,2,\ldots,n$ .

because

STATEMENT-2: $\displaystyle\sum_{i=1}^nP(H_i)=1$ .

Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1

Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1

Statement-1 is True, Statement-2 is False

Statement-1 is False, Statement-2 is True

Answer and worked solution

Official answer: D

Statement 1. By Bayes, $P(H_i\mid E)=\dfrac{P(E\mid H_i)P(H_i)}{P(E)}$ , and $P(E)<1$ , so $P(H_i\mid E)\ge P(E\mid H_i)P(H_i)$ , with equality when $P(E\cap H_i)=0$ (both sides then 0). So the strict inequality need not hold for every $i$ : False.

Statement 2. The events are mutually exclusive and exhaustive, so the probabilities add to 1: True.

(D)

Topics:Bayes' theorem

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IIT-JEE 2007 · Paper 1, Mathematics question 54