Question

A local county has an unemployment rate of 6.2%. A random sample of 18 employable people are picked at random from the county and are asked if they are employed. The distribution is a binomial. Round answers to 4 decimal places.

a) Find the probability that exactly 4 in the sample are unemployed.

b) Find the probability that there are fewer than 3 in the sample are unemployed.

c) Find the probability that there are more than 4 in the sample are unemployed.

d) Find the probability that there are at most 3 in the sample are unemployed.

Answer 1

p = 0.062

n = 18

P(X = x) = 18Cx * 0.062^x * (1 - 0.062)^18-x

a) P(X = 4) = 18C4 * 0.062⁴ * 0.938¹⁴ = 0.0185

b) P(X < 3) = P(X = 0) + P(X = 1) + P(X = 2)

                  = 18C0 * 0.062⁰ * 0.938¹⁸ + 18C1 * 0.062¹ * 0.938¹⁷ + 18C2 * 0.062² * 0.938¹⁶

                  = 0.9031

c) P(X > 4) = 1 - (P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4))

                  = 1 - (18C0 * 0.062⁰ * 0.938¹⁸ + 18C1 * 0.062¹ * 0.938¹⁷ + 18C2 * 0.062² * 0.938¹⁶ + 18C3 * 0.062³ * 0.938¹⁵ + 18C4 * 0.062⁴ * 0.938¹⁴ )

                  = 1 - 0.9960

                  = 0.0040

d) P(X < 3) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3)

                  = 18C0 * 0.062⁰ * 0.938¹⁸ + 18C1 * 0.062¹ * 0.938¹⁷ + 18C2 * 0.062² * 0.938¹⁶ + 18C3 * 0.062³ * 0.938¹⁵

                  = 0.9776