Respuesta :
Answer:
3.33% probability that both pens are defective.
Step-by-step explanation:
The pens are chosen without replacement, so we use the hypergeometric distribution to solve this question.
Hypergeometric distribution:
The probability of x sucesses is given by the following formula:
[tex]P(X = x) = h(x,N,n,k) = \frac{C_{k,x}*C_{N-k,n-x}}{C_{N,n}}[/tex]
In which:
x is the number of sucesses.
N is the size of the population.
n is the size of the sample.
k is the total number of desired outcomes.
Combinations formula:
[tex]C_{n,x}[/tex] is the number of different combinations of x objects from a set of n elements, given by the following formula.
[tex]C_{n,x} = \frac{n!}{x!(n-x)!}[/tex]
In this question:
2 defective, so x = 2.
25 pens, so N = 25.
Two pens will be selected, so n = 2.
5 are defective, so k = 5.
[tex]P(X = x) = h(x,N,n,k) = \frac{C_{k,x}*C_{N-k,n-x}}{C_{N,n}}[/tex]
[tex]P(X = 2) = h(2,25,2,5) = \frac{C_{5,2}*C_{20,0}}{C_{25,2}} = 0.0333[/tex]
3.33% probability that both pens are defective.
