InterviewSolution
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InterviewSolution
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If A, B and C are three events, then which of the following is incorrect ?A. P(Exatly two of A, B, C occur) `leP(A cap B)+P(B cap C)+P(C cap A)`B. `P(A cup B cup C) le P(A)+P(B)+P(C )`C. P(Exactly one of A, B, C occur) `le P(A)+P(B)+P(C )-P(B cap C)-P(C cap A)-P(A cap B)`D. P(A and at least one of B c, occurs)`ge P(A cap B)+P(A cap C)` |
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Answer» Correct Answer - D We have, P(Exactly two of A, B, C occur) `=P(A cap B cap overline(C ))+P(A cap overline(B) cap C)+P(overline(A) cap B cap C)` `=P(A cap B)-P(A cap B cap C)+P(A cap C)-P(A cap B cap C)+P(B cap C-P(A cap B cap C))` `=P(A cap B)+P(B cap C)+P(A cap C)-3P(A cap B cap C) le P(A cap B)+P(B cap C)+P(A cap C)` Also, `P(A cup B cup C)` `=P(A cup B)+P(C )-P{(A cup B) cap C}` `le P(A cup B)+P(C )`. `le P(A)+P(B)+P(C ) " " [becauseP(A cup B) le P(A)+P(B)]` Now, P(Exactly one of A, B, C occurs) `=P(A cap overline(B) cap overline(C ))+P(overline(A) cap overline(B) cap C)+P(overline(A) cap B cap overline(C ))` `=P(A cap overline(B cup C))+P(overline(A cup B)cap C)+P(B cap overline(A cup C))` `=P(A)-P{A cap (B cap )}+P(C )-P{C cap (A cup B)}+P(B)-P{B cap (A cup C)}` `=P(A)-P{(A cap B) cup (A cap C)+P(C )-P{(C cap A) cup (C cap B)}+P(B)-P{(B cap A) cap (B cap C)}` `=P(A)+P(B)+P(C )=2P(A cap B)-2P(B cap C)-2P(A cap C)+3P(A cap B cap C)` `[P(A)+P(B)+P(C )-P(A cap B)-P(A cap cap)]` `-P[(A)+P(B)+P(C )-P(A cap C)-3P(A cap B cap C)]` `=P(A)+P(B)+P(C )-P(A cap B)-P(B cap C)-P(A cap C)-P("Exactly two of A, B, C occur")` `le P(A)+P(B)+P(C )-P(A cap B)-P(B cap C)-P(A cap C)` Finally, P(A and atleast one of B, C occurs) `=P[A cap B cap C)]` `P[(A cap B) cup (A cap C)]` `=P(A cap B)+P(A cap C)-P(A cap B cap C)` `le P(A cap B)+P(A cap C)` So, option (d) is incorrect. |
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