NDA 2024 Maths · Mathematics · Probability · easy

Let A and B be events of a sample space with P(A)>0 and P(B)>0. 1. If A and B are independent, then P(A∩ B)=P(A)P(B). 2. If A and B are independent…

Single correct option

Let

A

and

B

be events of a sample space with

P(A)>0

and

P(B)>0

. Consider the statements:
1. If

A

and

B

are independent, then

P(A\cap B)=P(A)P(B)

.
2. If

A

and

B

are independent, then they cannot be mutually exclusive.
Which of the statements given above is/are correct?

A. 1 only
B. 2 only
C. Both 1 and 2
D. Neither 1 nor 2

Show answer and solution

Answer: (C) Both 1 and 2

Statement 1 is the definition of independence, so it is true. For statement 2, mutually exclusive events have

P(A\cap B)=0

, but independence would force

P(A\cap B)=P(A)P(B)>0

, a contradiction; hence independent events with positive probabilities cannot be mutually exclusive. Both are correct. Trap: confusing independence with mutual exclusiveness.

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