Question

(1 point) The rates of on-time flights for commercial jets are continuously tracked by the U.S. Department of Transportation. Recently, Southwest Air had the best reate with 80 % of its flights arriving on time. A test is conducted by randomly selecting 1212 Southwest flights and observing whether they arrive on time. (a) Find the probability that at least 1010 flights arrive late.

Answer 1

`P( X \geq 10)=P(X=10)+P(X=11)+P(X=12)`

`P(X=10)=(12C10)(0.8)^(10) (1-0.8)^(12-10)=0.2835`

`P(X=11)=(12C11)(0.8)^(11) (1-0.8)^(12-11)=0.2062`

`P(X=12)=(12C12)(0.8)^(12) (1-0.2)^(12-12)=0.0687`

`P( X \geq 10)=0.2835+0.2062+0.0687=0.5583`

Explanation:

Previous concepts

The binomial distribution is a "DISCRETE probability distribution that summarizes the probability that a value will take one of two independent values under a given set of parameters. The assumptions for the binomial distribution are that there is only one outcome for each trial, each trial has the same probability of success, and each trial is mutually exclusive, or independent of each other".

Solution to the problem

Let X the random variable of interest, on this case we now that:

`X \sim Binom(n=12, p=0.8)`

The probability mass function for the Binomial distribution is given as:

`P(X)=(nCx)(p)^x (1-p)^(n-x)`

Where (nCx) means combinatory and it's given by this formula:

`nCx=(n!)/((n-x)! x!)`

And we want to find this probability:

`P( X \geq 10)=P(X=10)+P(X=11)+P(X=12)`

`P(X=10)=(12C10)(0.8)^(10) (1-0.8)^(12-10)=0.2835`

`P(X=11)=(12C11)(0.8)^(11) (1-0.8)^(12-11)=0.2062`

`P(X=12)=(12C12)(0.8)^(12) (1-0.2)^(12-12)=0.0687`

`P( X \geq 10)=0.2835+0.2062+0.0687=0.5583`