Respuesta :
Answer:
0.323 = 32.3% probability that the director chooses 3 boy kittens and 5 girl kittens.
Step-by-step explanation:
The kittens are chosen 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:
[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 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:
[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]
A TV studio has brought in 8 boy kittens and 10 girl kittens for a cat food commercial.
This means that [tex]N = 8 + 10 = 18[/tex]
We want 3 boys, so [tex]k = 8[/tex]
The director is going to choose 8 of these kittens at random to be in the commercial.
This means that [tex]n = 8[/tex]
What is the probability that the director chooses 3 boy kittens and 5 girl kittens?
This is P(X = 3).
[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 = 3) = h(3,18,8,8) = \frac{C_{8,3}*C_{10,5}}{C_{18,8}} = 0.323[/tex]
0.323 = 32.3% probability that the director chooses 3 boy kittens and 5 girl kittens.
