JEE Advanced 2021 · Paper 1 · Mathematics Q13

← JEE Advanced 2021 · Paper 1

MathematicsMultiple correct+4 / -2Question 51 of 57

Let $E$ , $F$ and $G$ be three events having probabilities
$P(E) = \frac18,\quad P(F) = \frac16\quad\text{and}\quad P(G) = \frac14,\quad\text{and let } P(E\cap F\cap G) = \frac1{10}.$
For any event $H$ , if $H^c$ denotes its complement, then which of the following statements is (are) TRUE?

P(E\cap F\cap G^c) \le \frac1{40}

P(E^c\cap F\cap G) \le \frac1{15}

P(E\cup F\cup G) \le \frac{13}{24}

P(E^c\cap F^c\cap G^c) \le \frac5{12}

Answer and worked solution

Official answer: A, B, C

Check each option. Each pairwise intersection is at least $P(E\cap F\cap G)=\frac1{10}$ .

(A) $P(E\cap F\cap G^c)=P(E\cap F)-\frac1{10}\le\frac18-\frac1{10}=\frac1{40}$ . True.
(B) $P(E^c\cap F\cap G)\le P(F)-\frac1{10}=\frac1{15}$ . True.
(C) By inclusion-exclusion,
$P(E\cup F\cup G)\le\frac18+\frac16+\frac14-\frac3{10}+\frac1{10}=\frac{41}{120}\le\frac{13}{24}.$
True.
(D) $P(E^c\cap F^c\cap G^c)=1-P(E\cup F\cup G)\ge\frac{79}{120}>\frac5{12}$ . False.

(A), (B), (C)

Topics:Classical probability Sets and operations

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JEE Advanced 2021 · Paper 1, Mathematics question 13