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:
![P(X = x) = h(x,N,n,k) = \frac{C_{k,x}*C_{N-k,n-x}}{C_{N,n}}](https://tex.z-dn.net/?f=P%28X%20%3D%20x%29%20%3D%20h%28x%2CN%2Cn%2Ck%29%20%3D%20%5Cfrac%7BC_%7Bk%2Cx%7D%2AC_%7BN-k%2Cn-x%7D%7D%7BC_%7BN%2Cn%7D%7D)
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:
is the number of different combinations of x objects from a set of n elements, given by the following formula.
![C_{n,x} = \frac{n!}{x!(n-x)!}](https://tex.z-dn.net/?f=C_%7Bn%2Cx%7D%20%3D%20%5Cfrac%7Bn%21%7D%7Bx%21%28n-x%29%21%7D)
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.
![P(X = x) = h(x,N,n,k) = \frac{C_{k,x}*C_{N-k,n-x}}{C_{N,n}}](https://tex.z-dn.net/?f=P%28X%20%3D%20x%29%20%3D%20h%28x%2CN%2Cn%2Ck%29%20%3D%20%5Cfrac%7BC_%7Bk%2Cx%7D%2AC_%7BN-k%2Cn-x%7D%7D%7BC_%7BN%2Cn%7D%7D)
![P(X = 2) = h(2,25,2,5) = \frac{C_{5,2}*C_{20,0}}{C_{25,2}} = 0.0333](https://tex.z-dn.net/?f=P%28X%20%3D%202%29%20%3D%20h%282%2C25%2C2%2C5%29%20%3D%20%5Cfrac%7BC_%7B5%2C2%7D%2AC_%7B20%2C0%7D%7D%7BC_%7B25%2C2%7D%7D%20%3D%200.0333)
3.33% probability that both pens are defective.