A research team 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. At a board meeting of 15 businessmen, all of whom wear ties, what are the following probabilities?
a.) more than two ties are too tight
b.) no tie is too tight
To calculate the probabilities requested, we need to have some additional information. Specifically, we need to know the probability that a randomly chosen businessman who wears a tie wears it too tightly. Once we have that information, we can use the concept of the binomial distribution to calculate the probabilities.
For the purpose of this explanation, let's assume that the probability of a businessman wearing a tie too tightly is 0.10, as given in the study.
a.) Probability that more than two ties are too tight:
To calculate this probability, we can use the binomial distribution formula. In this case, we want to find the probability of more than two successes (i.e., businessmen wearing ties too tightly) in a sample of 15 businessmen. The formula is as follows:
P(X > 2) = 1 - P(X ≤ 2) = 1 - [P(X = 0) + P(X = 1) + P(X = 2)]
Using the binomial distribution formula, we can calculate each individual probability:
P(X = 0) = C(15, 0) * (0.10)^0 * (1 - 0.10)^(15 - 0)
P(X = 1) = C(15, 1) * (0.10)^1 * (1 - 0.10)^(15 - 1)
P(X = 2) = C(15, 2) * (0.10)^2 * (1 - 0.10)^(15 - 2)
Once we calculate these probabilities, we can substitute them into the equation and solve for P(X > 2).
b.) Probability that no tie is too tight:
To calculate this probability, we need to find the probability of having no successes (businessmen wearing ties too tightly) in a sample of 15 businessmen. Using the binomial distribution formula, we can calculate this probability:
P(X = 0) = C(15, 0) * (0.10)^0 * (1 - 0.10)^(15 - 0)
Once we calculate this probability, we get the probability that no tie is too tight.
Note: In real-world scenarios, the probability of businessmen wearing their ties too tightly might vary, and additional research may be needed to get accurate data for the calculation.