Question

A grocery store refrigerator contains 55 cartons of eggs. Of these, 40 cartons are fully intact while the rest contain at least one broken egg. A sample of 10 cartons will be selected (without replacement) from the 55 cartons in the refrigerator. Let X = number of cartons in the sample that are fully intact.

(a) The probability that the sample will contain exactly 8 cartons that are fully intact is .

(b) The expected value of X is .

(c) The variance of X is .

Answer 1

here number of cartons of eggs = N = 55

Here full intact cartons = K = 40

atleast one broken egg cartons = N - K = 55 - 40 = 15

sample size = n = 10

Here we are selecting it without replacement.

X = number of cartons in the sample that arefully intact.

(a) Here X have Hypergeometric distribution with parameter N =55, K =40 and n = 10

Pr(X = 8) = ⁴⁰C₈¹⁵C₂/⁵⁵C₁₀ = 0.2761

(b) Expected value of X is E[X] = (K/N) * n = (40/55) * 10 = 7.273

(c) VaR [X] = n* (K/N) * (N-K)/N * (N - n)/(N-1)

= 10 * (40/55) * (15/55) * (55 - 10)/(55 - 1) = 1.6529