Let and be three events having probabilities and let For any event , if denotes its complement, then which of the following statements is(are) TRUE? [2021]

Question

Let $E, F$ and $G$ be three events having probabilities $P (E) = \frac{1}{8}, P (F) = \frac{1}{6}, P (G) = \frac{1}{4},$ and let $P (E \cap F \cap G) = \frac{1}{10} .$

For any event $H$ , if $H^{c}$ denotes its complement, then which of the following statements is(are) TRUE? [2021]

Smart Achievers · Accepted Answer

(1, 2, 3)

$Given that$

$G$

(3) $P (E \cup F \cup G)$ $= P (E) + P (F) + P (G)$ $- P (E \cap F) - P (F \cap G) - P (G \cap E) + P (E \cap F \cap G)$

$= \frac{1}{8} + \frac{1}{6} + \frac{1}{4} - \sum P (E \cap F) + \frac{1}{10}$

$= \frac{13}{24} + \frac{1}{10} - \sum P (E \cap F)$

$\Rightarrow P (E \cup F \cup G) \leq \frac{13}{24}$ $[Option (3) is correct]$

(4) $Now, P (E^{c} \cap F^{c} \cap G^{c})$

$= 1 - P (E \cup F \cup G) \geq 1 - \frac{13}{24}$

$\Rightarrow P (E^{c} \cap F^{c} \cap G^{c}) \geq \frac{11}{24}$ $[Option (4) is incorrect]$

(1) $∵$ $P (E) \geq P (E \cap F \cap G) + P (E \cap F \cap G^{c})$

$\Rightarrow \frac{1}{8} \geq P (E \cap F \cap G^{c}) + \frac{1}{10}$

$\Rightarrow \frac{1}{8} - \frac{1}{10} \geq P (E \cap F \cap G^{c})$

$\Rightarrow \frac{1}{40} \geq P (E \cap F \cap G^{c})$ $[Option (1) is correct]$

(2) $∵$ $P (F) \geq P (E^{c} \cap F \cap G) + P (E \cap F \cap G)$

$\Rightarrow \frac{1}{6} \geq P (E^{c} \cap F \cap G) + \frac{1}{10}$

$\Rightarrow \frac{1}{6} - \frac{1}{10} \geq P (E^{c} \cap F \cap G)$

$\Rightarrow \frac{4}{60} \geq P (E^{c} \cap F \cap G)$

$\Rightarrow \frac{1}{15} \geq P (E^{c} \cap F \cap G)$ $[Option (2) is correct]$

Solution

Q.
Let $E, F$ and $G$ be three events having probabilities $P (E) = \frac{1}{8}, P (F) = \frac{1}{6}, P (G) = \frac{1}{4},$ and let $P (E \cap F \cap G) = \frac{1}{10} .$

For any event $H$ , if $H^{c}$ denotes its complement, then which of the following statements is(are) TRUE [2021]

1 $P (E \cap F \cap G^{c}) \leq \frac{1}{40}$

2 $P (E^{c} \cap F \cap G) \leq \frac{1}{15}$

3 $P (E \cup F \cup G) \leq \frac{13}{24}$

4 $P (E^{c} \cap F^{c} \cap G^{c}) \leq \frac{5}{12}$

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Solution

Q. Let E,F and G be three events having probabilities P(E)=18, P(F)=16, P(G)=14, and let P(E∩F∩G)=110. For any event H, if Hc denotes its complement, then which of the following statements is(are) TRUE [2021]

1 P(E∩F∩Gc)≤140

2 P(Ec∩F∩G)≤115

3 P(E∪F∪G)≤1324