#### 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?
- What is the expected number of “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 of exactly 50 “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?

#### WHAT TO SUBMIT

Create a report that includes an introduction paragraph, all of your solutions to the questions above, and a conclusion paragraph in a Word Document. Make sure to include all work and explanations along with your solutions. Proper formatting and use of correct symbols are required.