Please show all the work necessary to explain why your answer is correct!

1. Number of Combinations

Recall that you can compute the number of different ways to pick k items out of n possibilities (assuming that you are picking without replacement and that order does NOT matter) using the combination formula,

Ck(n)  = ✓ ◆k(n) ⌘ − k)!

(a) Prove that (x(n)) = (n x)

(b) Without using math, explain why this equality makes sense: Why should the number of ways you can draw x objects out of n be the same as the number of ways you can draw (n-x) objects out of n?

2. An Ultimate Frisbee team consists of six men and five women.  Three team members need to serve as Co- Captains and attend the League Meetings. The team agrees to randomly choose the three Co-Captains.

(a) How many different groups of Co-Captains are possible?

(b) What is the probability that there will be no women Co-Captains?

(c) What is the probability that there will be exactly two male Co-Captains and exactly one female Co- Captain?

3.  Suppose that an instructor announces that the next test will consist of eight questions, which will be randomly selected from a list of twenty questions handed out one week before the exam. In order to make a C or better, a student must be able to answer at least five out of the eight test questions selected. Going into the test, Mackenzie knows the correct answers to only ten of the twenty questions. The questions are such that the answers cannot be guessed.

(a) What is the probability that Mackenzie will get a perfect score?

(b) What is the probability that Mackenzie will make a C or better?

4.  Suppose that of all the students taking macrotheory at Brandeis this semester, 0% are freshmen, 25% are sopho- mores, 60% are juniors and 15% are seniors. Suppose that sophomores are 95% likely to turn in problem sets, juniors are 80% likely to turn in problem sets, and seniors are 75% likely to turn in problem sets.

(a)  Overall, what fraction of macrotheory problem sets will be turned in this semester?

(b) Construct a table with joint and marginal probabilities:

Let the rows be “turn problem sets in” and “do not turn problem sets in”

Let the columns be “freshmen”, “sophomores”, “juniors” and “seniors”

Fill in all the numbers you can.