Question

According to Harper's Index, 50% of all federal inmates are serving time for drug dealing. A random sample of 15 federal inmates is selected.

(a) What is the probability that 12 or more are serving time for drug dealing? (Round your answer to three decimal places.)


(b) What is the probability that 6 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? (Round your answer to one decimal place.)

Answer 1

This is a binomial distribution

n = 15

p = 0.5

P(X = x) = 15Cx * 0.5^x * (1 - 0.5)^15-x

a) P(X > 12) = P(X = 12) + P(X = 13) + P(X = 14) + P(X = 15)

                     = 15C12 * 0.5¹² * 0.5³ + 15C13 * 0.5¹³ * 0.5² + 15C14 * 0.5¹⁴ * 0.5¹ + 15C15 * 0.5¹⁵ * 0.5⁰

                     = 0.018

b) P(X < 6) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) + P(X = 5) + P(X = 6)

                   = 15C0 * 0.5⁰ * 0.5¹⁵ + 15C1 * 0.5¹ * 0.5¹⁴ + 15C2 * 0.5² * 0.5¹³ + 15C3 * 0.5³ * 0.5¹² + 15C4 * 0.5⁴ * 0.5¹² + 15C5 * 0.5⁵ * 0.5¹⁰ + 15C6 * 0.5⁶ * 0.5⁹

                   = 0.304

c) Expected value = n * p = 15 * 0.5 = 7.5