Respuesta :
Answer:
0.294 = 29.4% probability that none of the chosen entrees contain spinach.
Step-by-step explanation:
The choices of entrees are chosen without replacement, which means that the hypergeomtric 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]
13 different entrees for an important banquet
This means that [tex]N = 13[/tex]
3 of which contain spinach.
This means that [tex]k = 3[/tex]
The guests will have a choice of 4 entrees.
This means that [tex]n = 4[/tex]
What is the probability that none of the chosen entrees contain spinach?
This is P(X = 0). So
[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(0,13,4,3) = \frac{C_{3,0}*C_{10,4}}{C_{13,4}} = 0.294[/tex]
0.294 = 29.4% probability that none of the chosen entrees contain spinach.
