Answer :
Answer:
[tex]P(X > 0) = 0.9222[/tex]
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.
[tex]P(X = x) = C_{n,x}.\pi^{x}.(1-\pi)^{n-x}[/tex]
In which [tex]C_{n,x}[/tex] is the number of different combinatios 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 [tex]\pi[/tex] is the probability of X happening.
In this problem.
5 names are selected, so [tex]n = 5[/tex]
A success is a name being non-authentic. 40% of the names are non-authentic, so [tex]\pi = 0.40[/tex].
We have to find [tex]P(X > 0)[/tex]
Either the number of non-authentic names is 0, or is greater than 0. The sum of these probabilities is decimal 1. So:
[tex]P(X = 0) + P(X > 0) = 1[/tex]
[tex]P(X > 0) = 1 - P(X = 0)[/tex]
[tex]P(X = x) = C_{n,x}.\pi^{x}.(1-\pi)^{n-x}[/tex]
[tex]P(X = 0) = C_{5,0}*(0.40)^{0}*(0.6)^{5} = 0.0778[/tex]
So
[tex]P(X > 0) = 1 - P(X = 0) = 1-0.0778 = 0.9222[/tex]