Question

Given two independent events, if the probability that exactly one of them occurs is `26/49` and the probability that none of them occurs is `15/49`, then the probability of more probable of the two events is ______.

Options

• `4/7`

• `6/7`

• `3/7`

• `5/7`

Answer 1

Given two independent events, if the probability that exactly one of them occurs is `26/49` and the probability that none of them occurs is `15/49`, then the probability of more probable of the two events is `underlinebb(4/7)`.

Explanation:

Let the probability of occurrence of first event A, be ‘a’

i.e., P(A) = a `\implies` P(not A) = 1 – a

And also suppose that the probability of occurrence of second event B, P(B) = b,

∴ P(not B) = 1 – b

Now, P(A and not B) + P(not A and B) = `26/49`

`\implies` P(A) × P(not B) + P(not A) × P(B) = `26/49`

`\implies` a × (1 – b) + (1 – a) b = `26/49`

`\implies` a + b – 2ab = `26/49`   ...(i)

And P(not A and not B) = `15/49`

`\implies` P(not A) × P(not B) = `15/49`

`\implies` (1 – a) × (1 – b) = `15/49`

`\implies` 1 – b – a + ab = `15/49`

`\implies` a + b – ab = `34/49`  ...(ii)

From (i) and (ii),

a + b = `42/49` and ab = `8/49`  ...(iii)

(a – b)² = (a + b)² – 4ab = `42/49 xx 42/49 - (4 xx 8)/49 = 196/2401`

∴ a – b = `14/49`  ...(iv)

From (iii) and (iv),

a = `4/7`, b = `2/7`

Hence the probability of more probable of the two events = `4/7`