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 (1) `1/((log)_(10)^4-(log)_(10)^3)`
(2) `1/((log)_(10)^4+(log)_(10)^3)`
(3) `9/((log)_(10)^4-(log)_(10)^3)`
(4) `4/((log)_(10)^4-(log)_(10)^3)`

Options

A. `(1)/(log_(10)4-log_(10)3)`
B. `(1)/(log_(10)4+log_(10)3)`
C. `(9)/(log_(10)4-log_(10)3)`
D. `(4)/(log_(10)4-log_(10)3)`

Answer

A

Explanation

Let X be the binomial variate. Then,
`P(Xge 1)ge (9)/(10)`
`rArr1-P(X=0)gt(9)/(10)`
`rArr P(X=0)lt (1)/(10)`
`rArr .^Nc_(0)((1)/(4))^0((3)/(4))^n lt (1)/(10)`
`rArr n log_(10)((3)/(4))lt log ((1)/(10))rArr n (log_(10)4-log_(10)3)lt 0log_(10)10`
`rArr n(log_10 4-log_(10)3)gt 1rArr n gt (1)/(log_(10)4-log_(10)10)`