Answer:
0.0667 = 6.67% probability that all seven machines are nondefective.
Step-by-step explanation:
The machines are chosen from the sample 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:
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.
In this question:
10 machines means that ![n = 10](https://tex.z-dn.net/?f=n%20%3D%2010)
2 defective, so 10 - 2 = 8 work correctly, which means that ![k = 8](https://tex.z-dn.net/?f=k%20%3D%208)
Seven are selected, which means that ![n = 7](https://tex.z-dn.net/?f=n%20%3D%207)
What is the probability that all seven machines are nondefective?
This is P(X = 7). 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 = 7) = h(7,10,7,8) = \frac{C_{8,7}*C_{2,0}}{C_{10,7}} = 0.0667](https://tex.z-dn.net/?f=P%28X%20%3D%207%29%20%3D%20h%287%2C10%2C7%2C8%29%20%3D%20%5Cfrac%7BC_%7B8%2C7%7D%2AC_%7B2%2C0%7D%7D%7BC_%7B10%2C7%7D%7D%20%3D%200.0667)
0.0667 = 6.67% probability that all seven machines are nondefective.