Respuesta :
Answer:
a) P(X = 0) = 0.5223
b) P(X > 2) = 0.0125
Step-by-step explanation:
A probability is the number of desired outcomes divided by the number of total outcomes.
The order that the items are selected is not important, so the combinations formula is used to solve this problem.
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, we have that
100 items
6 defective
94 not defective.
a) P{X=0}
None defective.
Desired outcomes
Combinations of 10 from a set of 94. So
[tex]D = C_{94,10} = \frac{94!}{10!84!} = 9041256800000[/tex]
Total outcomes:
Combinations of 10 from a set of 100. So
[tex]T = C_{100,10} = \frac{100!}{10!90!} = 17310309000000[/tex]
P(X = 0)
[tex]p = \frac{D}{T} = \frac{9041256800000}{17310309000000} = 0.5223[/tex]
(b) P(X>2}.
Either two or less are defective, or more than two are defective. The sum of the probabilities of these events is decimal 1. So
[tex]P(X \leq 2) + P(X >2) = 1[/tex]
[tex]P(X > 2) = 1 - P(X \leq 2)[/tex]
In which
[tex]P(X \leq 2) = P(X = 0) + P(X = 1) + P(X = 2)[/tex]
P(X = 0)
[tex]P(X = 0) = \frac{D}{T} = \frac{9041256800000}{17310309000000} = 0.5223[/tex]
P(X = 1)
Desired outcomes
Combinations of 9 from a set of 94(non defecive) and 1 from a set of 6(defective). So
[tex]D = C_{84,9}*C_{6,1} = \frac{94!}{9!85!}*\frac{6!}{1!5!} = 6382063700000[/tex]
Total outcomes:
Combinations of 10 from a set of 100. So
[tex]T = C_{100,10} = \frac{100!}{10!90!} = 17310309000000[/tex]
P(X = 1)
[tex]P(X = 1) = \frac{D}{T} = \frac{6382063700000}{17310309000000} = 0.3687[/tex]
P(X = 2)
Desired outcomes
Combinations of 8 from a set of 94(non defecive) and 2 from a set of 6(defective). So
[tex]D = C_{94,8}*C_{6,2} = \frac{94!}{8!86!}*\frac{6!}{2!4!} = 1669726000000[/tex]
Total outcomes:
Combinations of 10 from a set of 100. So
[tex]T = C_{100,10} = \frac{100!}{10!90!} = 17310309000000[/tex]
P(X = 2)
[tex]P(X = 2) = \frac{D}{T} = \frac{1669726000000}{17310309000000} = 0.0965[/tex]
Finally
[tex]P(X \leq 2) = P(X = 0) + P(X = 1) + P(X = 2) = 0.5223 + 0.3687 + 0.0965 = 0.9875[/tex]
[tex]P(X > 2) = 1 - P(X \leq 2) = 1 - 0.9875 = 0.0125[/tex]
