Answer:
0.0013 probability that at least 6 employees were over 50.
Step-by-step explanation:
The employees were "chosen" to be dismissed without replacement, which means that the hypergeometric distribution is used 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 problem:
8 employees dismissed means that ![n = 8](https://tex.z-dn.net/?f=n%20%3D%208)
Had 7 + 17 = 24 employees, which means that ![N = 24](https://tex.z-dn.net/?f=N%20%3D%2024)
7 over 50, which means that ![k = 7](https://tex.z-dn.net/?f=k%20%3D%207)
What is the probability that at least 6 employees were over 50?
6 or 7, so:
.
In which
![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 = 6) = h(6,24,8,7) = \frac{C_{7,6}*C_{17,2}}{C_{24,8}} = 0.0013](https://tex.z-dn.net/?f=P%28X%20%3D%206%29%20%3D%20h%286%2C24%2C8%2C7%29%20%3D%20%5Cfrac%7BC_%7B7%2C6%7D%2AC_%7B17%2C2%7D%7D%7BC_%7B24%2C8%7D%7D%20%3D%200.0013)
![P(X = 7) = h(7,24,8,7) = \frac{C_{7,7}*C_{17,1}}{C_{24,8}} \approx 0](https://tex.z-dn.net/?f=P%28X%20%3D%207%29%20%3D%20h%287%2C24%2C8%2C7%29%20%3D%20%5Cfrac%7BC_%7B7%2C7%7D%2AC_%7B17%2C1%7D%7D%7BC_%7B24%2C8%7D%7D%20%5Capprox%200)
![P(X \geq 6) = P(X = 6) + P(X = 7) = 0.0013 + 0 = 0.0013](https://tex.z-dn.net/?f=P%28X%20%5Cgeq%206%29%20%3D%20P%28X%20%3D%206%29%20%2B%20P%28X%20%3D%207%29%20%3D%200.0013%20%2B%200%20%3D%200.0013)
0.0013 probability that at least 6 employees were over 50.