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 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

To calculate the probabilities, we need to use the concept of binomial distribution. The probability of at least one tie being too tight is equal to 1 minus the probability that no tie is too tight. The probability of more than two ties being too tight can be calculated by finding the sum of the probability of 3 ties being too tight, 4 ties being too tight, and so on, up to 15 ties being too tight. The probability of no tie being too tight is the complement of the probability of at least one tie being too tight. Lastly, the probability of at least 13 ties not being too tight is equal to the probability of 13 ties not being too tight, plus the probability of 14 ties not being too tight, and the probability of 15 ties not being too tight.

Step-by-step explanation:

To calculate the probabilities, we need to use the concept of binomial distribution. The probability of at least one tie being too tight is equal to 1 minus the probability that no tie is too tight. The probability of more than two ties being too tight can be calculated by finding the sum of the probability of 3 ties being too tight, 4 ties being too tight, and so on, up to 15 ties being too tight. The probability of no tie being too tight is the complement of the probability of at least one tie being too tight. Lastly, the probability of at least 13 ties not being too tight is equal to the probability of 13 ties not being too tight, plus the probability of 14 ties not being too tight, and the probability of 15 ties not being too tight.

1. (a) P(at least one tie is too tight) = 1 - P(no tie is too tight)
2. (b) P(more than two ties are too tight) = P(3 ties are too tight) + P(4 ties are too tight) + ... + P(15 ties are too tight)
3. (c) P(no tie is too tight)
4. (d) P(at least 13 ties are not too tight) = P(13 ties are not too tight) + P(14 ties are not too tight) + P(15 ties are not too tight)