shipment of 30 computers, 8 are defective. Four computers are chosen at random. Find the probability that:
a)Exactly two are defective
b)None are defective
c)At most two are defective
a-1/4
b-2/30
c-3/4
To solve these problems, we first need to calculate the total number of possible outcomes and the number of favorable outcomes for each scenario.
Let's start by calculating the total number of possible outcomes:
Since we are choosing 4 computers out of 30, the total number of possible outcomes is the number of ways to choose 4 computers from a set of 30. This can be calculated using the combination formula:
Total possible outcomes = C(30, 4) = 30! / (4! * (30-4)!) = 27,405
Now let's calculate the number of favorable outcomes for each scenario:
a) Exactly two are defective:
To have exactly two defective computers, we need to choose two defective computers out of the 8 defective ones, and we also need to choose two working computers out of the 22 non-defective ones.
Number of favorable outcomes = C(8, 2) * C(22, 2) = (8! / (2! * (8-2)!)) * (22! / (2! * (22-2)!)) = 2,772
b) None are defective:
To have none of the four chosen computers be defective, we need to choose four working computers out of the 22 non-defective ones.
Number of favorable outcomes = C(22, 4) = 22! / (4! * (22-4)!) = 7315
c) At most two are defective:
To find the number of favorable outcomes where at most two computers are defective, we need to sum up the number of favorable outcomes for having 0, 1, or 2 defective computers.
Number of favorable outcomes = C(8, 0) * C(22, 4) + C(8, 1) * C(22, 3) + C(8, 2) * C(22, 2) = (1) * (7315) + (8) * (7315) + (2,772) = 27,897
Now that we have calculated the number of favorable outcomes for each scenario, we can calculate the probability:
a) Probability of exactly two defective computers = Number of favorable outcomes / Total possible outcomes = 2,772 / 27,405 ≈ 0.101
b) Probability of none of the four computers being defective = Number of favorable outcomes / Total possible outcomes = 7,315 / 27,405 ≈ 0.267
c) Probability of at most two defective computers = Number of favorable outcomes / Total possible outcomes = 27,897 / 27,405 ≈ 1.018
Note: The probabilities in parts a) and b) are approximate values rounded to three decimal places.