Question

In a binomial distribution `B(n, p = 1/4)`, if the probability of at least one success is greater than or equal to `9/10`, then n is greater than ______.

Options

• `1/(log_10 4 + log_10 3)`

• `9/(log_10 4 - log_10 3)`

• `4/(log_10 4 - log_10 3)`

• `1/(log_10 4 - log_10 3)`

Answer 1

In a binomial distribution `B(n, p = 1/4)`, if the probability of at least one success is greater than or equal to `9/10`, then n is greater than `underlinebb(1/(log_10 4 - log_10 3))`

Explanation:

Given that P = `1/4` `\implies` q = `1 - 1/4 = 3/4`

and `P(x ≥ 1) ≥ 9/10 \implies 1 - P(x = 0) ≥ 9/10`

`\implies 1 - ""^nC_0 (1/4)^0 (3/4)^n ≥ 9/10`

`\implies 1 - 9/10 ≥ (3/4)^n`

`\implies (3/4)^n ≤ (1/10)`

Taking log at the base `3/4`, on both sides, we get

`n log_(3/4) ≥ log_(3/4) (1/10)`

`\implies n ≥ - log_(3/4) 10 = (- log_10 10)/(log_10 (3/4)) = (-1)/(log_10 3 - log_10 4)`

`\implies n ≥ 1/(log_10 4 - log_10 3)`