Answer:
31.82% probability that you will receive 2 apples.
Step-by-step explanation:
The fruits are removed from the bag, 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 question:
5 + 7 = 12 total fruits, which means that ![N = 12](https://tex.z-dn.net/?f=N%20%3D%2012)
7 apples, which means that ![k = 7](https://tex.z-dn.net/?f=k%20%3D%207)
You receive 2 fruits, which means that ![n = 2](https://tex.z-dn.net/?f=n%20%3D%202)
What is the probability, in percent, that you will receive 2 apples, assuming she removes them from the bag in random order?
This is, as a proportion, P(X = 2). 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 = 2) = h(2,12,2,7) = \frac{C_{7,2}*C_{5,0}}{C_{12,2}} = 0.3182](https://tex.z-dn.net/?f=P%28X%20%3D%202%29%20%3D%20h%282%2C12%2C2%2C7%29%20%3D%20%5Cfrac%7BC_%7B7%2C2%7D%2AC_%7B5%2C0%7D%7D%7BC_%7B12%2C2%7D%7D%20%3D%200.3182)
0.3182*100% = 31.82%
31.82% probability that you will receive 2 apples.