Question

Let E₁ and E₂ be two independent events. Let P(E) denotes the probability of the occurrence of the event E. Further, let E'₁ and E'₂ denote the complements of E₁ and E₂, respectively. If P(E'₁ ∩ E₂) = `2/15` and P(E₁ ∩ E'₂) = `1/6`, then P(E₁) is

Options

• `2/15`

• `13/15`

• `2/13`

• `1/5`

Answer 1

`1/5`

Explanation:

Let P(E₁) = m ⇒ P(E₁¹) = 1 – m

Given that , P(E₁¹ ∩ E₂) = P(E₁¹) . P(E₂) = `2/15`

⇒ (1 – m)P(E₂) = `2/15` ⇒ P(E₂) = `2/(15(1 - m))`

and P(E₁ ∩ E₂¹) = `1/6` ⇒ P(E₁) . P(E₂¹) = `1/6`

⇒ `m(1 - 2/(15(1 - m))) = 1/6`

⇒ `6m [15(1 - m) - 2] = 15(1 - m)`

⇒ `2m(13 - 15 m) = 5 - 5m`

⇒ `30m^2 - 31m + 5` = 0

⇒ `m = 5/6` or `1/5`

⇒ P(E₁) = `5/6` or `1/5`.