Suppose in a box containing 30 laptops, 3 are defective. You randomly select 10 laptops out of the box. What is the probability that 7 of the 10 selected laptops are good. What is the type of this probability distribution?
Question
Suppose in a box containing 30 laptops, 3 are defective. You randomly select 10 laptops out of the box. What is the probability that 7 of the 10 selected laptops are good. What is the type of this probability distribution?
Solution
This is a hypergeometric distribution problem. The hypergeometric distribution describes the probability of k successes (random draws for which the object drawn has a specified feature) in n draws, without replacement, from a finite population of size N that contains exactly K objects with that feature.
The formula for the hypergeometric probability is:
P(X=k) = [C(K, k) * C(N-K, n-k)] / C(N, n)
where:
- P(X=k) is the probability of k successes in n draws
- C(N, n) is the number of combinations of N items taken n at a time
- C(K, k) is the number of combinations of K successes taken k at a time
- C(N-K, n-k) is the number of combinations of N-K failures taken n-k at a time
- N is the total number of items
- K is the total number of successes
- n is the number of draws
- k is the number of successes in n draws
In this case:
- N = 30 (the total number of laptops)
- K = 30 - 3 = 27 (the total number of good laptops)
- n = 10 (the number of laptops selected)
- k = 7 (the number of good laptops selected)
Substituting these values into the formula, we get:
P(X=7) = [C(27, 7) * C(30-27, 10-7)] / C(30, 10)
Calculating the above expression gives a probability of approximately 0.3504 when rounded to four decimal places.
So, the probability that 7 of the 10 selected laptops are good is 0.3504. The type of this probability distribution is hypergeometric.
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