QJEEAdv21Mae72b: JEE Advanced 2021 Mathematics - Class 12 JEEAdv21Mae72b Solution: Let E, F and G be three events having probabilities , and , and …

Question

Verified Solution • 2025–26 Session moderate

Let E, F and G be three events having probabilities

P(E) = {1 \over 8}

,

P(F) = {1 \over 6}

and

P(G) = {1 \over 4}

, and let P (E

\cap

F

\cap

G) =

{1 \over {10}}

. For any event H, if H^c denotes the complement, then which of the following statements is (are) TRUE?

Accepted Answer

For option (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)

= {1 \over 8} - {1 \over {10}} = {1 \over {40}}

For option (b)

P({E^C} \cap F \cap G) = P(F \cap G) - P(E \cap F \cap G) \le P(F) - P(E \cap F \cap G)

= {1 \over 6} - {1 \over {10}} = {1 \over {15}}

For option (c)

P(E \cup F \cup G) \le P(E) + P(F) + P(G)

= {1 \over 8} + {1 \over 6} + {1 \over 4} = {{3 + 4 + 6} \over {24}} = {{13} \over {24}}

For option (d)

P({E^C} \cap {F^C} \cap {G^C}) = 1 - P(E \cap F \cap G) \ge 1 - {{13} \over {24}}

P({E^C} \cap {F^C} \cap {G^C}) \ge {{11} \over {24}} &gt; {5 \over {12}}

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