Respuesta :
Answer:
0.8 = 80% probability that either your aunt or your father will become a city commissioner.
Step-by-step explanation:
The candidates 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:
[tex]P(X = x) = h(x,N,n,k) = \frac{C_{k,x}*C_{N-k,n-x}}{C_{N,n}}[/tex]
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:
[tex]C_{n,x}[/tex] is the number of different combinations of x objects from a set of n elements, given by the following formula.
[tex]C_{n,x} = \frac{n!}{x!(n-x)!}[/tex]
In this question:
6 candidates, which means that [tex]N = 6[/tex]
2 are the aunt and father, which means that [tex]k = 2[/tex]
3 are chosen, which means that [tex]n = 3[/tex]
What is the probability that either your aunt or your father will become a city commissioner?
Probability of at least one of them being chosen, which is:
[tex]P(X \geq 1) = 1 - P(X = 0)[/tex]
In which
[tex]P(X = x) = h(x,N,n,k) = \frac{C_{k,x}*C_{N-k,n-x}}{C_{N,n}}[/tex]
[tex]P(X = 0) = h(0,6,3,2) = \frac{C_{2,0}*C_{4,3}}{C_{6,3}} = 0.2[/tex]
Then
[tex]P(X \geq 1) = 1 - P(X = 0) = 1 - 0.2 = 0.8[/tex]
0.8 = 80% probability that either your aunt or your father will become a city commissioner.
