Question

Suppose that you own a car dealership and purchase (randomly select) 10 cars of a certain make from a production run of 200 cars. Of the 200 cars, 160 are destined to last at least five years without needing a major repair.

1. Using the hypergeometric distribution what is the probability that at least 6 of your 10 cars will last at least five years without needing a major repair.

Answer 1

Hypergeometric Distribution .

Suppose a population consists of N items, k of which are successes. And a random sample drawn from that population consists on n items, x of which are successes. Then the hypergeometric probability is:

For the Given Problem,

Population : N : Number of cars in the production run = 200

Number of successes in the population : k : Number of cars that are destined to last at least five years without needing a major repair = 160

Number of items in the sample : n : Number of cars purchased(chosen randomly) = 10

X: Number of successes of the sample : Number of cars in the sample that are destined to last at least five years without needing a major repair

N-k = 200-160 = 40

Probability that at least 6 cars out 10 will last at least five years without needing a major repair = P(X6)

P(X6) = P(X=6) + P(X=7) + P(X=8) + P(X=9) + P(X=10)

P(X=6) + P(X=7) + P(X=8) + P(X=9) + P(X=10) = 0.08627+0.20518+0.30980+0.26832+0.10129=0.97086

Probability that at least 6 cars out 10 will last at least five years without needing a major repair = 0.97086