Question

A research team conducted a study showing that approximately 25% 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

a. 0.967

b. 0.909

c. 0.0032

d. 0.0913

Explanation:

Please see the attached for a detailed explanation

Answer 2

Answer:A) 0.9866

Explanation:

Probability of at least one tie is too tight is

Probability of tight tie 25/100

Probability of not a tight tie is 1-25/100=3/4

=1-(3/4)^15=0.9866

B)0.764

C)0.013

D)0.236

Method 2

This is binomial probability.

n=15 (number of businessmen

p=.25 (probability that a businessman wears the tie so tightly

A)

x = number of businessmen out of 15 who wears the tie so tightly

P( at least 1) = P( x >= 1) = 1-P(x=0)

P(x=0) = 15C0 (.25)^0 (.75)^(15-0) = (0.75)^15 = 0.013363

P( at least 1) = 1- 0.013363 = 0.9866

B)

P(more than 2) = P( x > 2) = 1-P( x=0)-P(x=1)-P(x=2)

P(x=0) = 15C0 (.25)^0 (.75)^(15-0) = (0.75)^15 = 0.013363

P(x=1) = 15C1 (.25)^1 (.75)^14 = 0.066817

P(x=2) = 15C2 (.25)^2 (.75)^13 = 0.155907

1-P( x=0)-P(x=1)-P(x=2) = 1 - 0.013363 - 0.066817 - 0.155907 = 0.763912

C)

P(x=0) = 15C0 (.25)^0 (.75)^(15-0) = (0.75)^15 = 0.013363

D)

P( at least 13) = P(x=13)+P(x=14)+P(x=15)

P(x=13)= 15C13 (.25)^13 (.75)^2 = 0.000001

P(x=14)= 15C14 (.25)^14 (.75)^1 = 0.000000

P(x=15)= 15C15 (.25)^15 (.75)^0 = 0.000000

P(x=13)+P(x=14)+P(x=15) = .000001+.000000+.000000 = 0.000001