Question

Dorothy Little purchased a mailing list of 2,000 names and addresses for her mail order business, but after scanning the list she doubts the authenticity of the list. She randomly selects five names from the list for validation. If 40% of the names on the list are non-authentic, and x is the number of non-authentic names in her sample, P(x>0) is ______________.

Answer 1

Answer:

$P(X > 0) = 0.9222$

Step-by-step explanation:

For each name, there are only two outcomes. Either the name is authentic, or it is not. So, we can solve this problem using the binomial probability distribution.

Binomial probability distribution

The binomial probability is the probability of exactly x successes on n repeated trials, and X can only have two outcomes.

$P(X = x) = C_{n,x}.\pi^{x}.(1-\pi)^{n-x}$

In which $C_{n,x}$ is the number of different combinatios of x objects from a set of n elements, given by the following formula.

$C_{n,x} = \frac{n!}{x!(n-x)!}$

And $\pi$ is the probability of X happening.

In this problem.

5 names are selected, so $n = 5$

A success is a name being non-authentic. 40% of the names are non-authentic, so $\pi = 0.40$.

We have to find $P(X > 0)$

Either the number of non-authentic names is 0, or is greater than 0. The sum of these probabilities is decimal 1. So:

$P(X = 0) + P(X > 0) = 1$

$P(X > 0) = 1 - P(X = 0)$

$P(X = x) = C_{n,x}.\pi^{x}.(1-\pi)^{n-x}$

$P(X = 0) = C_{5,0}*(0.40)^{0}*(0.6)^{5} = 0.0778$

So

$P(X > 0) = 1 - P(X = 0) = 1-0.0778 = 0.9222$