Question

At Denver International Airport, 83% of recent flights have arrived on time. A sample of 12 flights is studied. (a) Calculate the probability that all 12 flights were on time

Answer 1

10.69% probability that all 12 flights were on time

Explanation:

For each flight, there are only two possible outcomes. Either it was on time, or it was not. The probability of a flight being on time is independent of any other flight. So we use the binomial probability distribution to solve this question.

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.

83% of recent flights have arrived on time.

This means that
`p = 0.83`

A sample of 12 flights is studied.

This means that
`n = 12`

Calculate the probability that all 12 flights were on time

This is P(X = 12).

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

`P(X = 12) = C_(12,12).(0.83)^(12).(0.17)^(0) = 0.1069`

10.69% probability that all 12 flights were on time