Question

A research team conducted a study showing that approximately 20% of all businessmen who wear ties wear them so tightly that they actually reduce blood flow to the brain, diminishing cerebral functions. At a board meeting of 20 businessmen, all of whom wear ties, what are the following probabilities? (Round your answers to three decimal places.)

(a) at least one tie is too tight


(b) more than two ties are too tight


(c) no tie is too tight


(d) at least 18 ties are not too tight

Answer 1

p = 0.2

n = 20

This is a binomial distribution

P(X = x) = 20Cx * 0.2^x * (1 - 0.2)^20-x

a) P(X > 1) = 1 - P(X = 0) = 1 - 20C0 * 0.2⁰ * 0.8²⁰ = 1 - 0.0115 = 0.9885

b) P(X > 2) = 1 - P(X < 2)

                  = 1 - (P(X = 0) + P(X = 1) + P(X = 2))

                  = 1 - (20C0 * 0.2⁰ * 0.8²⁰ + 20C1 * 0.2¹ * 0.8¹⁹ + 20C2 * 0.2² * 0.8¹⁸ )

                  = 1 - 0.2061

                  = 0.7939

c) P(X = 0) = 20C0 * 0.2⁰ * 0.8²⁰ = 0.011

d) (at least 18 not too tight) = P(at most 2 too tight) = P(X < 2) = 20C0 * 0.2⁰ * 0.8²⁰ + 20C1 * 0.2¹ * 0.8¹⁹ + 20C2 * 0.2² * 0.8¹⁸ = 0.2061