Question

A manufactured lot of buggy whips has 20 items, of which 5 are defective. A random sample of 5 items is chosen to be inspected. Find the probability that the sample contains exactly one defective item

Answer 1

`P(X=1)`

And using the probability mass function we got:

`P(X=1) = (5C1) (0.25)^1 (1-0.25)^(5-1)= 0.396`

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

For this cae that one buggy whip would be defective is
`p = (5)/(20)=0.25`

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

`X \sim Binom(n=5, p=0.25)`

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=1)`

And using the probability mass function we got:

`P(X=1) = (5C1) (0.25)^1 (1-0.25)^(5-1)= 0.396`