Let E, F and G be three events having probabilities PE=18, PF=16 and PG=14, and let PE∩F∩G=110For any event H, if HC denotes its complement, then which of the following statements is (are) TRUE ?

P E C ∩ F C ∩ G C ≤ 5 12 , P E ∩ F ∩ G C ≤ 1 40 , P E ∪ F ∪ G ≤ 13 24

Let E, F and G be three events having probabilities PE=18, PF=16 and PG=14, and let PE∩F∩G=110For any event H, if HC denotes its complement, then which of the following statements is (are) TRUE ?

P E = 1 8 , P F = 1 6 , P G = 1 4 , P E ∩ F ∩ G − 1 10 A P E ∩ F ∩ G C = P E ∩ F − P E ∩ F ∩ G ≤ P E − P E ∩ F ∩ G ≤ 1 8 − 1 10 ≤ 5 − 4 40 ≤ 1 40 B P E C ∩ F ∩ G = P F ∩ G − P E ∩ F ∩ G ≤ P F − P E ∩ F ∩ G ≤ 1 6 − 1 10 ≤ 10 − 6 60 ≤ 4 60 ≤ 1 15 C P E ∪ F ∪ G ≤ P E + P E + P G ≤ 1 8 + 1 6 + 1 4 ≤ 13 24 D P E C ∪ F C ∪ G C = 1 − P E ∪ F ∪ G ≥ 1 − 13 24 ≥ 11 24 > 10 24 > 5 12

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Let E, F and G be three events having probabilities
$P (E) = \frac{1}{8}, P (F) = \frac{1}{6} a n d P (G) = \frac{1}{4}, a n d l e t P (E \cap F \cap G) = \frac{1}{10}$
For any event H, if $H^{C}$ denotes its complement, then which of the following statements is (are) TRUE ?

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

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

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

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

answer is A, B, C.

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Detailed Solution

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

$(A) P (E \cap F \cap G^{C}) = P (E \cap F) - P (E \cap F \cap G) \leq P (E) - P (E \cap F \cap G) \leq \frac{1}{8} - \frac{1}{10} \leq \frac{5 - 4}{40} \leq \frac{1}{40}$

$(B) P (E^{C} \cap F \cap G) = P (F \cap G) - P (E \cap F \cap G) \leq P (F) - P (E \cap F \cap G) \leq \frac{1}{6} - \frac{1}{10} \leq \frac{10 - 6}{60} \leq \frac{4}{60} \leq \frac{1}{15} (C) P (E \cup F \cup G) \leq P (E) + P (E) + P (G) \leq \frac{1}{8} + \frac{1}{6} + \frac{1}{4} \leq \frac{13}{24} (D) P (E^{C} \cup F^{C} \cup G^{C}) = 1 - P (E \cup F \cup G) \geq 1 - \frac{13}{24} \geq \frac{11}{24} > \frac{10}{24} > \frac{5}{12}$