Probability of binomial

Question

According to Harper's Index, 40% of all federal inmates are serving time for drug dealing. A random sample of 20 federal inmates is selected.

(a) What is the probability that 13 or more are serving time for drug dealing? (Round your answer to three decimal places.)

(b) What is the probability that 7 or fewer are serving time for drug dealing? (Round your answer to three decimal places.)

(c) What is the expected number of inmates serving time for drug dealing?

Answer 1

np = 20 * 0.4 = 8 nq = 20 * 0.6 = 12

since both are greater than 5 can use normal approximation to binomial

mean = np = 8

standard deviation = sqrt(npq) = 2.19

standardized normal variable z = (x - mean)/SD, where x is number serving time

P(x >= 13) = P(x > 12.5) using continuity correction

P(x > 12.5) = P(z > (12.5 - 8)/2.19) = P(z > 2.05) = P(z < -2.05) = 0.0202

P(x <= 7) = P(x < 7.5) = P(z < (7.5 - 8)/2.19) = P(z < -0.23) = 0.4090

expected value = np = 8