Stats Problem Please Help

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 15 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 13 ties are not too tight

Answer 1

This is called a binary probability problem. There are two possibilities for each individual: tie too tight or tie not too tight. The probability of a person’s tie being too tight is .20, but we need to find the probability of various outcomes in a group of these individuals. 

The formula for x number of people having ties too tight:

(n!/(n-x)!) p^(x) q^(n-x)

Where p=.20, q=1-p=.80, and n=15

We need to apply this formula in specific ways to get the answers for the various scenarios. 

a) For P(at least 1), it’s easiest to calculate P(0) and subtract that from 1. P(at least 1) means every case other than the one where no one has a tie that’s too tight. If your using a graphing calculator or computer program that can calculate cumulative probabilities, you probably don’t care about this shortcut. When you’re working by hand or if you eventually get a problem with less information provided, it’s good to know how to put this information to work for you! :)

So for part a, use the above formula with x=0 and then subtract your answer from 1. 

1-P(0)

b) Calculate P(1) and P(2), then subtract those from your answer from part a. 

1-P(0)-P(1)-P(2)

c) You have already found this answer in part a. So use P(0) from part a as your answer. 

P(0)

d) At least 13 ties are not too tight means that 2 or fewer are too tight. So you just need to add P(0), P(1), and P(2) together. 

P(0)+P(1)+P(2)

You can check your answers to this problem (and any similar problems) by using an online binomial probability calculator. Please let me know if you have any other questions, and up vote this if you found it helpful. Thanks!

Answer 2

Hi Been,

These sound like binomal distribution probability problems. Visit https://www.geogebra.org/classic#probability for example, and change the model to binomal. Make n equal to the number of businessmen 15, and p equal to the probability .2. Adjust the values you want the random variable X to be, either by typing or by dragging the black triangles on the graph.

Tim