Answer:
0.9167 = 91.67% probability that the entire batch will be accepted
Step-by-step explanation:
The CDs are chosen without replacement, which means that the hypergeometric distribution is used.
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)
2100 write-rewrite CDs, and 60 are defective.
This means, respectively, that ![N = 2100, k = 60](https://tex.z-dn.net/?f=N%20%3D%202100%2C%20k%20%3D%2060)
3 of these CDs are randomly selected for testing
This means that ![n = 3](https://tex.z-dn.net/?f=n%20%3D%203)
What is the probability that the entire batch will be accepted?
Probability of none defective, so P(X = 0).
![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 = 0) = h(0,2100,3,60) = \frac{C_{60,0}*C_{2040,3}}{C_{2100,3}} = 0.9167](https://tex.z-dn.net/?f=P%28X%20%3D%200%29%20%3D%20h%280%2C2100%2C3%2C60%29%20%3D%20%5Cfrac%7BC_%7B60%2C0%7D%2AC_%7B2040%2C3%7D%7D%7BC_%7B2100%2C3%7D%7D%20%3D%200.9167)
0.9167 = 91.67% probability that the entire batch will be accepted