Respuesta :
Answer:
B) 0.74545
Step-by-step explanation:
A probability is the number of desired outcomes divided by the number of total outcomes.
The hypergeometric distribution can be simplified using the combinations formula.
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 problem:
12 cans
3 contain an incorrect mix.
4 are selected
Probability that at least one of the four cans selected contains an incorrect mix of paint.
Either none contain an incorrect mix, or at least one does. The sum of the probabilities of these events is decimal 1. So
[tex]P(X = 0) + P(X \geq 1) = 1[/tex]
We want [tex]P(X \geq 1)[/tex]
So
[tex]P(X \geq 1) = 1 - P(X = 0)[/tex]
Finding P(X = 0):
None with an incorrect mix
Desired outcomes:
4 from a set of 9(without an incorrect mix). So
[tex]D = C_{9,4} = \frac{9!}{4!(9-4)!} = 126[/tex]
Total outcomes:
4 from a set of 12. So
[tex]T = C_{12,4} = \frac{12!}{4!(12-4)!} = 495[/tex]
Probability:
[tex]P(X = 0) = p = \frac{D}{T} = \frac{126}{495} = 0.25455[/tex]
Probability that at least one of the four cans selected contains an incorrect mix of paint:
[tex]P(X \geq 1) = 1 - P(X = 0) = 1 - 0.25455 = 0.74545[/tex]
So the correct answer is:
B) 0.74545
