These problems are mostly from Blitzstein and Hwang, with slight modifications.

Remember: It’s very important that you work on all of these problems using your own brain!

Sets

1. Let \(A\) and \(B\) be events of some sample space. Write down the event which means ‘\(A\) and \(B\) or neither \(A\) nor \(B\)’

2. Let \(A_n\) be the set of positive integers that are divisible by \(n\). What is the smallest element of \(A_2 \cap A_3\)?

Counting

3. 1. How many 7-digit phone numbers are possible if the first digit can’t be a 0 or a 1?

   2. Redo (a), but now also assume that the phone number cannot start with ‘911’.

4. Fred is planning to go out to dinner each night of a certain week, Monday through Friday, with each dinner being at one of his ten favorite restaurants.

   1. How many possibilities are there for Fred’s schedule of dinners for that Monday through Friday, if Fred is not willing to eat at the same restaurant more than once?
   2. How many possibilities are there for Fred’s schedule of dinners for that Monday through Friday, if Fred is willing to eat at the same restaurant more than once, but is not willing to eat at the same place twice in a row (or more)?

5. To fulfill the requirements for a certain degree, a student can choose to take any 7 out of a list of 20 courses, with the constraint that at least 1 of the 7 courses must be a statistics course. Suppose that 5 of the 20 courses are statistics courses.

   1. How many choices are there for which 7 courses to take?
   2. Is this the same as \({5 \choose 1} \times {19 \choose 6}\)? Why or why not?

6. How many ways are there to permute the letters in the word MISSISSIPPI? (Swapping two of the same letters doesn’t count as a different permutation.)

7. A round-robin tournament is being held with \(n\) tennis players; this means that every player will play against every other player exactly once.

   1. How many games are played in total?

   2. How many possible outcomes are there for the tournament? (An outcome is a list of who won and lost every game.)

8. March Madness is a single-elimination basketball tournament, meaning that every team plays one game per round, the loser of a game is eliminated, and the last remaining team is the champion.

   Pretend there are exactly \(2^n\) teams in the first round. (In reality, some teams get a bye, so the number of teams isn’t exactly a power of 2.)

   1. How many rounds are there?
   2. Count how many games in total are played by adding up the numbers of games played in each round.
   3. Count how many games in total are played by thinking about it, with almost no calculation. Hint: How many players need to be eliminated?

Probability by counting

9. Three people get into an empty elevator at the first floor of a building that has 10 floors. Each presses the button for their desired floor. Assume that they are equally likely to want to go to floors 2 through 10 (independently of each other). What is the probability that the buttons for 3 consecutive floors are pressed?

10. A certain family has 6 children, consisting of 3 boys and 3 girls. Assuming that all birth orders are equally likely, what is the probability that the 3 eldest children are the 3 girls?

11. A city with 6 districts has 6 robberies in a particular week. Assume that a robber chooses his target randomly, with all possibilities equally likely. What is the probability that some district had more than 1 robbery?

12. A survey of 1000 people is being conducted in a city with 1 million residents. The 1000 people are chosen with one at a time, with replacement, with equal probabilities. Find the probability that at least one person will get chosen more than once.

13. A college has 10 time slots for its courses, and blithely assigns courses to completely random time slots, independently. The college offers exactly 3 statistics courses. What is the probability that 2 or more of the statistics courses are in the same time slot?

14. A random 5-card poker hand is dealt from a standard deck of cards. Find the probability of each of the following possibilities in terms of binomial coefficients. If you’re brave, compute them numerically.

    1. Two pair.
    2. A flush (all five cards are of the same suit)
    3. A royal flush (meaning that the ranks are exactly ace, king, queen, jack, and 10).

15. Robert, Natasha, and ten other people sit at a round table by randomly choosing seats. Find the probability that Robert is sitting next to Natasha in two ways:

    1. Using a sample space with \(12!\) elements, where each outcome says where everyone is sitting.

    2. Using a much smaller sample space that ignores the ten other people. (There are a few correct answers here.)

Inclusion-exclusion

2026-04-02: This is moved to next week’s homework! I’ll leave it here so you can see it.

16. A club consists of 10 seniors, 12 juniors, and 15 sophomores. An organizing committee of five people is chosen randomly, meaning all subsets of size 5 are equally likely.

    1. Find the probability that there are exactly 3 sophomores on the committee. (This doesn’t require inclusion-exclusion.)

    2. Find the probability that the committee has at least one senior, junior, and sophomore.

17. What is the probability that 13 random cards from a 52-card deck are void in at least one suit? (That means that there exists one suit which is not contained in the hand.)

18. A fair, six-sided die is rolled \(n\) times. This question is about the probability that you see every value at least once.

    1. Obviously, as \(n\) gets bigger, you would expect to see every value. I will bet you $100 that you don’t see every value in \(n\) rolls. Gut feeling, do you feel comfortable with \(n = 10\)?

    2. Find a formula for the probability that at least one of the six possible values never appears.

    3. How do you feel about \(n = 10\) now? What’s the smallest \(n\) that you like?