Question

Fifteen cars are inspected for California emissions compliance. Twelve were found to be compliant and 3 were not. A sample of 7 cars is chosen at random. What is the probability that all 7 cars chosen are compliant?

Answer 1

Answer:

$P(X=7)$

$P(X=k)= \frac{(MCk)(N-M C n-k)}{NCn}$  

Where N=15 is the population size, M=12 is the number of success states in the population, n=7 is the number of draws, k is the number of observed successes  

And using the probability mass function we got:

$P(X=7)= \frac{(12C7)(15-12 C 7-7)}{15C7}=\frac{792*1}{6435}=0.123$

So then the answer for this case would be 0.123.

Step-by-step explanation:

Previous concepts

The hypergeometric distribution is a discrete probability distribution that its useful when we have more than two distinguishable groups in a sample and the probability mass function is given by:

$P(X=k)= \frac{(MCk)(N-M C n-k)}{NCn}$  

Where N=15 is the population size, M=12 is the number of success states in the population, n=7 is the number of draws, k is the number of observed successes  

The expected value and variance for this distribution are given by:

$E(X)= n\frac{M}{N}$

$Var(X)=n \frac{M}{N}\frac{N-M}{N}\frac{N-n}{N-1}$

Solution to the problem

For this case we want this probability:

$P(X=7)$

And using the probability mass function we got:

$P(X=7)= \frac{(12C7)(15-12 C 7-7)}{15C7}=\frac{792*1}{6435}=0.123$

So then the answer for this case would be 0.123.