Question

Each egg in a carton of 12 eggs has a 0.1 probability of being broken, independent of all the other eggs. Suppose I purchase 20 cartons (each containing twelve eggs). What is the probability that there will be exactly 15 out of 20 cartons that will contain some broken eggs?  

Answer 1

Here, n = 12, p = 0.1, (1 - p) = 0.9 and x = 0
As per binomial distribution formula P(X = x) = nCx * p^x * (1 - p)^(n - x)

We need to calculate P(X = 0)
P(X = 0) = 12C0 * 0.1^0 * 0.9^12
P(X = 0) = 0.2824

Hence the probability that at least one egg is broken in the carton is 1 - 0.2824 = 0.7176

Now,
Here, n = 20, p = 0.7176, (1 - p) = 0.2824 and x = 15
As per binomial distribution formula P(X = x) = nCx * p^x * (1 - p)^(n - x)

We need to calculate P(X = 15)
P(X = 15) = 20C15 * 0.7176^15 * 0.2824^5
P(X = 15) = 0.1919

Ans: 0.1919