If  and , then which one of the following is not correct?

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NDA (Held On: 6 Sep 2020) Maths Previous Year paper
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  1. P(A ∩ B) = P(A) P(B)
  2. P(A ∪ B) > P(A) + P(B)
  3. P(not A and not B) = P(not A) P(not B)

Answer (Detailed Solution Below)

Option 3 : P(A ∪ B) > P(A) + P(B)
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NDA 01/2025: English Subject Test
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Detailed Solution

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Concept:

Complement of an event:

The complement of an event is the subset of outcomes in the sample space that are not in the event.

The probability of the complement of an event is one minus the probability of the event.

P (A̅) = 1 - P (A) ⇒ P (A) = 1 - P (A̅)

Formula: P (A ∪ B) =  P (A) + P (B) - P (A ∩ B)

 

Calculation: 

Given: 

So, P(A) = 1 - P (A̅) = 

As we know, P (A ∪ B) =  P (A) + P (B) - P (A ∩ B)

⇒ P (A ∪ B) =  1 - P (A̅) + P (B) - P (A ∩ B)

Hence option 1 is true.

 

P(A ∩ B) = P(A) P(B) = 

Hence option 2 is also true.

 

P(A) + P(B) =  >  P(A ∪ B) 

Hence option 3 is not true.

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