Question

Let E₁ and E₂ be the events such that P(E₁) = 1/3 and P(E₂) = 3/5.

Find:

(i) P(E₁∪ E₂), when E₁ and E₂ are mutually exclusive.

(ii) P(E₁∩ E₂), when E₁ and E₂ are independent

Answer 1

Given: E₁ and E₂ are two events such that P(E₁) = \(\frac{1}{3}\) and P(E₂) = \(\frac{3}{5}\)

To Find: 

(i) P(E₁ ∪ E₂) when E₁ and E₂ are mutually exclusive.

We know that,

When two events are mutually exclusive P(E₁ ∩ E₂) = 0

Hence, P(E₁ ∪ E₂) = P(E₁) + P(E₂)

= \(\frac{1}{3}+\frac{3}{5}\)

= \(\frac{14}{15}\)

Therefore , P(E₁ ∪ E₂) = \(\frac{14}{15}\) when E₁ and E₂ are mutually exclusive.

(ii) P(E₁ ∩ E₂) when E₁ and E₂ are independent.

We know that when E₁ and E₂ are independent ,

P(E₁ ∩ E₂) = P(E₁) x P(E₂)

= \(\frac{1}{3}\times \frac{3}{5}\)

= \(\frac{1}{5}\)

Therefore, P(E₁ ∩ E₂) = \(\frac{1}{5}\) when E₁ and E₂ are independent.