Scenario

The dating service, Hearts and Diamonds, specializes in matching people who enjoy entertainment venues. The company has a number of counting problems involved with their business model. As part of the client intake process, clients are asked to answer eighty "yes/no questions," twenty "values" questions where the answers ranged from 1 10, a question asking the client to rank eight "hard choices" ordinally, that is, by first, second, third, etc., and a free-form question about ten "likes and dislikes.

" Assume that every client has a 60% chance of answering each "yes/no" question with a "yes," and that the answer to each "yes/no" question is independent of every other answer to the other "yes/no" questions.

Instructions Based on the information given above, answer the following questions.

What is the probability that two different clients will have the same answer to question #2 (a "yes/no" question)?

What is the expected number of "yes" answers to "yes/no" questions? What is the probability of exactly 50 "yes" answers to "yes/no" questions?

What is the probability that a client selects at most 40 "yes" answers to "yes/no" questions?

What is the probability that the hard choice, "I am late to work," is ranked #1 and the hard choice, "I get no coffee this week," is ranked #8 if all rankings are equally likely?

What is the probability that Barbara, Carl, Georg, and Henrietta are randomly chosen from a set of twenty clients for a four-person group date if all possible choices are equally likely?