Question

17. Northern Airlines has a reputation for being on time 86% of the time.

Find the probability that a Northern Airlines plane will be on time for

at least four of its next five flights.

Answer 1

85.33% probability that a Northern Airlines plane will be on time for at least four of its next five flights.

Explanation:

For each Northern Airlines flight, there are only two possible outcomes. Either it is on time, or it is not on time. The probability of a flight being on time is independent of other flights. 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.

Northern Airlines has a reputation for being on time 86% of the time.

This means that
`p = 0.86`

Find the probability that a Northern Airlines plane will be on time for at least four of its next five flights.

`P(X \geq 4) = P(X = 4) + P(X = 5)`

In which

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

`P(X = 4) = C_(5,4).(0.86)^(4).(0.14)^(1) = 0.3829`

`P(X = 4) = C_(5,4).(0.86)^(5).(0.14)^(0) = 0.4704`

`P(X \geq 4) = P(X = 4) + P(X = 5) = 0.3829 + 0.4704 = 0.8533`

85.33% probability that a Northern Airlines plane will be on time for at least four of its next five flights.