JEE Advanced 2021 Paper 1 Q13 Mathematics P&C and Probability Probability Medium

JEE Advanced 2021 Paper 1 · Q13 · Probability

Let $E, F$ and $G$ be three events having probabilities $P(E) = \dfrac{1}{8}$, $P(F) = \dfrac{1}{6}$ and $P(G) = \dfrac{1}{4}$, and let $P(E \cap F \cap G) = \dfrac{1}{10}$. For any event $H$, if $H^c$ denotes its complement, then which of the following statements is (are) TRUE ?

  1. A. $P(E \cap F \cap G^c) \leq \dfrac{1}{40}$
  2. B. $P(E^c \cap F \cap G) \leq \dfrac{1}{15}$
  3. C. $P(E \cup F \cup G) \leq \dfrac{13}{24}$
  4. D. $P(E^c \cap F^c \cap G^c) \leq \dfrac{5}{12}$
JEE Advanced multi-correct — pick every correct option, then check.
Reveal answer + step-by-step solution

Correct answer:A, B, C

Solution

(A) $P(E\cap F\cap G^c)=P(E\cap F)-P(E\cap F\cap G)\le P(E)-P(E\cap F\cap G)=\dfrac{1}{8}-\dfrac{1}{10}=\dfrac{1}{40}$. TRUE. (B) $P(E^c\cap F\cap G)=P(F\cap G)-P(E\cap F\cap G)\le P(F)-\dfrac{1}{10}=\dfrac{1}{6}-\dfrac{1}{10}=\dfrac{4}{60}=\dfrac{1}{15}$. TRUE. (C) $P(E\cup F\cup G)\le P(E)+P(F)+P(G)=\dfrac{1}{8}+\dfrac{1}{6}+\dfrac{1}{4}=\dfrac{3+4+6}{24}=\dfrac{13}{24}$. TRUE. (D) $P(E^c\cap F^c\cap G^c)=1-P(E\cup F\cup G)\ge 1-\dfrac{13}{24}=\dfrac{11}{24}$, which is not $\le \dfrac{5}{12}=\dfrac{10}{24}$. FALSE.

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