Question

1. Thirty percent of the employees of a large company are minorities. A random sample of 6 employees is selected.

1. What is the probability that the sample contains exactly 4 minorities?

2.   What is the probability that the sample contains fewer than 3 minorities?
3. What is the expected number of minorities in the sample?

Answer 1

a)

Here, n = 6, p = 0.3, (1 - p) = 0.7 and x = 4
As per binomial distribution formula P(X = x) = nCx * p^x * (1 - p)^(n - x)

We need to calculate P(X = 4)
P(X = 4) = 6C4 * 0.3^4 * 0.7^2
P(X = 4) = 0.0595

b)

Here, n = 6, p = 0.3, (1 - p) = 0.7 and x = 3
As per binomial distribution formula P(X = x) = nCx * p^x * (1 - p)^(n - x)

We need to calculate P(X <3).
P(X < 3) = (6C0 * 0.3^0 * 0.7^6) + (6C1 * 0.3^1 * 0.7^5) + (6C2 * 0.3^2 * 0.7^4)
P(X < 3) = 0.1176 + 0.3025 + 0.3241
P(X <3) = 0.7442

c)

Expected number = np
= 6 * 0.30
= 1.8