At a certain airport, 75% of the flights arrive on time. A sample of 10 flights is studied. Assume each flight is independent of the others. Find the probability that eight or more of the flights will arrive on time.

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

[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]

In which [tex]C_{n,x}[/tex] is the number of different combinations of x objects from a set of n elements, given by the following formula.

[tex]C_{n,x} = \frac{n!}{x!(n-x)!}[/tex]

And p is the probability of X happening.

At a certain airport, 75% of the flights arrive on time.

This means that [tex]p = 0.75[/tex]

A sample of 10 flights is studied.

This means that [tex]n = 10[/tex]

Find the probability that eight or more of the flights will arrive on time.

[tex]P(X \geq 8) = P(X = 8) + P(X = 9) + P(X = 10)[/tex]

In which

[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]

[tex]P(X = 8) = C_{10,8}.(0.75)^{8}.(0.25)^{2} = 0.2816[/tex]

[tex]P(X = 9) = C_{10,9}.(0.75)^{9}.(0.25)^{1} = 0.1877[/tex]

[tex]P(X = 10) = C_{10,10}.(0.75)^{10}.(0.25)^{0} = 0.0563[/tex]

[tex]P(X \geq 8) = P(X = 8) + P(X = 9) + P(X = 10) = 0.2816 + 0.1877 + 0.0563 = 0.5256[/tex]

52.56% probability that eight or more of the flights will arrive on time.

MrRoyal MrRoyal · Answer 2 · 2022-01-13T13:37:40+01:00

The probability that eight or more of the flights will arrive on time is 0.5256

Probabilities

Probabilities are used to determine the chances of events

The given parameters are:

Sample size: [tex]n = 10[/tex]
The proportion of flights that arrive on time [tex]p = 0.75[/tex].

The required probability is then represented as:

[tex]P(x \ge 8)[/tex]

Binomial probability

This is calculated using the following binomial probability formula

[tex]P(X = x) = ^nC_x p^x (1 - p)^{n -x}[/tex]

So, we have:

[tex]P(x \ge 8) =P(x = 8) + P(x = 9) + P(x =10)[/tex]

This gives

[tex]P(x \ge 8) = ^{10}C_8 \times 0.75^8 \times 0.25^2 +^{10}C_9 \times 0.75^9 \times 0.25 +^{10}C_{10} \times 0.75^{10} \times 0.25^0[/tex]

This gives

[tex]P(x \ge 8) = 0.5256[/tex]

Hence, the probability is 0.5256