Answer:
0.3968 = 39.68% probability this shipment passes inspection.
Step-by-step explanation:
The parts are chosen without replacement, which means that the hypergeometric distribution is used to solve this question.
Hypergeometric distribution:
The probability of x successes 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 successes.
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:
50 parts means that ![N = 50](https://tex.z-dn.net/?f=N%20%3D%2050)
4 defective means that ![k = 4](https://tex.z-dn.net/?f=k%20%3D%204)
10 are chosen, which means that ![n = 10](https://tex.z-dn.net/?f=n%20%3D%2010)
What is the probability this shipment passes inspection?
Probability that none is defective, so:
![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(0,50,10,4) = \frac{C_{4,0}*C_{46,10}}{C_{50,10}} = 0.3968](https://tex.z-dn.net/?f=P%28X%20%3D%200%29%20%3D%20h%280%2C50%2C10%2C4%29%20%3D%20%5Cfrac%7BC_%7B4%2C0%7D%2AC_%7B46%2C10%7D%7D%7BC_%7B50%2C10%7D%7D%20%3D%200.3968)
0.3968 = 39.68% probability this shipment passes inspection.