Question

A study showed that approximately 10% of all businessman who wear ties wear them so tightly that they actually reduce blood flow to the brain. At a board meeting of 20 businessman, all of the businessmen are wearing ties. what is the probability that: a) What is the probability that cxactly two of the businessmen have ties that are too tight? Do by hand, using the formulas. b(Verify your answer in a) using the appropriate calculator distr/DISTR tool. c) What is the probability that more than two of the businessmen have ties that are too tight? Use the appropriate calculator distr/DISTR tool.

Answer 1

The probability of exactly two businessmen wearing ties too tight is calculated using the binomial probability formula and verified with a calculator's distr/DISTR tool. The probability of more than two ties being too tight is found by subtracting the cumulative probability of two or fewer from one.

Step-by-step explanation:

The probability that exactly two businessmen have ties that are too tight can be calculated using the binomial probability formula:

P(X = x) = C(n, x) * p^x * (1 - p)^(n - x)

Where:

Here, n = 20, x = 2, and p = 0.10. Plugging these values in, we get:

P(X = 2) = C(20, 2) * 0.10^2 * (1 - 0.10)^(20 - 2)

C(20, 2) = 20! / (2! * (20 - 2)!) = 190

P(X = 2) = 190 * 0.10^2 * 0.90^18

P(X = 2) = 0.2852 (rounded to four decimal places)

To verify the answer using a calculator's distribution tool, one would use the binomial distribution function: binompdf(n, p, x) where n is the number of trials, p is the probability of success, and x is the number of successes.

For the probability that more than two businessmen's ties are too tight, we use the calculator's cumulative distribution function, minus the probabilities that none, one, or two ties are too tight:

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

Where P(X = 0) and P(X = 1) can be found using the calculator's binompdf function, and P(X = 2) is the probability we calculated manually.