Question

The mean and variance of a binomial distribution are α and `α/3` respectively. If P(X = 1) = `4/243`, then P(X = 4 or 5) is equal to ______.

Options

• `5/9`

• `64/81`

• `16/27`

• `145/243`

Answer 1

The mean and variance of a binomial distribution are α and `α/3` respectively. If P(X = 1) = `4/243`, then P(X = 4 or 5) is equal to `underlinebb(16/27)`.

Explanation:

Given mean = α, variance = `α/3` and

P(x = 1) = `4/243`

`overline"X"` = nP = α  ...(i)

σ² = npq = `α/3`  ...[From (i)]

`\implies` αq = `α/3`

`\implies` q = `1/3`

`\implies` P = 1 – q = `2/3`

Here, P(X = 1) = `4/243`

`""^"n""C"_1(2/3)^1(1/3)^("n" - 1) = 4/243`

`"n"(2/3)(1/3^("n" - 1)) = 4/243` = n = 6

So, P(X = 4 or X = 5) = `""^6"C"_4(2/3)^4(1/3)^2 + ""^6"C"_5(2/3)^5(1/3)^1`

= `(2/3)^4(1/3) xx 9`

= `16/27`