Question

If \(\rm P(A\cup B)=\dfrac{5}{6}, P(A\cap B)=\dfrac{1}{3}\:and\:P(\bar A)=\dfrac{1}{2}\), then which of the following is/are correct?

1. A and B are independent events.

2. A and B are mutually exclusive events.

Select the correct answer using the code given below.

1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2

Answer 1

Correct Answer - Option 1 : 1 only

Concept:

For any events A and B

• P(A ∪ B) = P(A) + P(B) - P(A ∩ B)
• Events are independent if P(A ∩ B) = P(A) × P(B)
• Events are mutually exclusive if P(A ∩ B) = 0
• P(\(\rm\bar A\)) = 1 - P(A)

 

Calculation:

Given \(\rm P(A\cup B)=\dfrac{5}{6}, P(A∩ B)=\dfrac{1}{3}\:and\:P(\bar A)=\dfrac{1}{2}\)

P(A) = 1 - \(1\over2\) = \(1\over2\)

P(A ∪ B) = P(A) + P(B) - P(A ∩ B)

\(\rm{5\over6} = {1\over2}+P(B)-{1\over3}\)

P(B) = \(2\over3\)

Given P(A ∩ B) ≠ 0

P(A ∩ B) = P(A) × P(B) = \({1\over2}\times{2\over3} = {1\over3}\)

∴ A and B are independent events and are not mutually exclusive events.

Only statement 1 is correct.