Question

Assume the random variable x has a binomial distribution with the given probability of obtaining a success. Find the following. Probabilitygiven the number of trials and the probability of obtaining a success. Round your answer to four decimal places. P(x>10), n=14, p= .8

Answer 1

P(x > 10) = 0.6981.

Explanation:

Binomial probability distribution

The binomial probability is the probability of exactly x successes on n repeated trials, and X can only have two outcomes.

`P(X = x) = C_(n,x).p^(x).(1-p)^(n-x)`

In which
`C_(n,x)` is the number of different combinations of x objects from a set of n elements, given by the following formula.

`C_(n,x) = (n!)/(x!(n-x)!)`

And p is the probability of X happening.

In this question:

`n = 14, p = 0.8`

P(x>10)

`P(x > 10) = P(X = 11) + P(X = 12) + P(X = 13) + P(X = 14)`

In which

`P(X = x) = C_(n,x).p^(x).(1-p)^(n-x)`

`P(X = 11) = C_(14,11).(0.8)^(11).(0.2)^(3) = 0.2501`

`P(X = 12) = C_(14,12).(0.8)^(12).(0.2)^(2) = 0.2501`

`P(X = 13) = C_(14,13).(0.8)^(13).(0.2)^(1) = 0.1539`

`P(X = 14) = C_(14,14).(0.8)^(14).(0.2)^(0) = 0.0440`

`P(x > 10) = P(X = 11) + P(X = 12) + P(X = 13) + P(X = 14) = 0.2501 + 0.2501 + 0.1539 + 0.0440 = 0.6981`

So P(x > 10) = 0.6981.