Question

A research team conducted a study showing that approximately 15% 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

Solution:

We are given n = 20, p = 0.15

Part a

We have to find P(X≥1)

P(X≥1) = 1 – P(X=0)

P(X=x) = nCx*p^x*(1 – p)^(n – x)

P(X=0) = 20C0*0.15^0*(1 – 0.15)^(20 – 0)

P(X=0) = 1*1*0.85^20

P(X=0) = 0.03876

P(X≥1) = 1 – P(X=0)

P(X≥1) = 1 – 0.03876

P(X≥1) = 0.96124

Required probability = 0.961

Part b

Here, we have to find P(X>2)

P(X>2) = 1 – P(X≤2)

P(X≤2) = P(X=0) + P(X=1) + P(X=2)

From above part, we have P(X=0) = 0.03876

P(X=x) = nCx*p^x*(1 – p)^(n – x)

P(X=1) = 20C1*0.15^1*(1 – 0.15)^(20 – 1) = 0.136798

P(X=2) = 20C2*0.15^2*(1 – 0.15)^(20 – 2) = 0.229338

P(X≤2) = 0.03876 + 0.136798 + 0.229338

P(X≤2) = 0.404896

P(X>2) = 1 – P(X≤2)

P(X>2) = 1 – 0.404896

P(X>2) = 10.595104

Required probability = 0.595

Part c

Here, we have to find P(X=0)

P(X=x) = nCx*p^x*(1 – p)^(n – x)

P(X=0) = 20C0*0.15^0*(1 – 0.15)^(20 – 0)

P(X=0) = 1*1*0.85^20

P(X=0) = 0.03876

Required probability = 0.039

Part d

Here, we have to find P(X≥18) = 1 – P(X≤17)

We are given n = 20, p = 0.15

So, by using binomial table or excel, we have

P(X≤17) = 0.99999999

P(X≥18) = 1 – P(X≤17)

P(X≥18) = 1 - 0.99999999

P(X≥18) = 0.00000001

Required probability = 0.000