Question

A research team at Cornell University conducted a study showing that approximately 10% of all businessmen who wear ties wear them so tightly that they actually reduce blood flow to the brain, diminishing cerebral functions (Source: Chances: Risk and Odds in Everyday Life, by James Burke). At a board meeting of 20 businessmen, all of whom wear ties, what is the probability that:

a) at least 5 ties are too tight?
b) at most 12 ties are too tight?

Answer 1

a) The probability that at least 5 ties are too tight is P=0.0432.

b) The probability that at most 12 ties are too tight is P=1.

Explanation:

In this problem, we could represent the proabilities of this events with the Binomial distirbution, with parameter p=0.1 and sample size n=20.

a) We can express the probability that at least 5 ties are too tight as:

`P(x\geq5)=1-\sum\limits^4_(k=0) {(n!)/(k!(n-k)!) p^k(1-p)^(n-k)}\\\\P(x\geq5)=1-(0.1216+0.2702+0.2852+0.1901+0.0898)\\\\P(x\geq5)=1-0.9568=0.0432`

The probability that at least 5 ties are too tight is P=0.0432.

a) We can express the probability that at most 12 ties are too tight as:

`P(x\leq 12)=\sum\limits^(12)_(k=0) {(n!)/(k!(n-k)!) p^k(1-p)^(n-k)}\\\\P(x\leq 12)=0.1216+0.2702+0.2852+0.1901+0.0898+0.0319+0.0089+0.0020+0.0004+0.0001+0.0000+0.0000+0.0000\\\\P(x\leq 12)=1`

The probability that at most 12 ties are too tight is P=1.