Randomized team drafting strategy

This Riddler Classic puzzle explores a randomized team drafting strategy designed to prevent teams from throwing games.

You are one of 30 team owners in a professional sports league. In the past, your league set the order for its annual draft using the teams’ records from the previous season — the team with the worst record got the first draft pick, the team with the second-worst record got the next pick, and so on. However, due to concerns about teams intentionally losing games to improve their picks, the league adopts a modified system. This year, each team tosses a coin. All the teams that call their coin toss correctly go into Group A, and the teams that lost the toss go into Group B. All the Group A teams pick before all the Group B teams; within each group, picks are ordered in the traditional way, from worst record to best. If your team would have picked 10th in the old system, what is your expected draft position under the new system?

Extra credit: Suppose each team is randomly assigned to one of T groups where all the teams in Group 1 pick, then all the teams in Group 2, and so on. (The coin-flipping scenario above is the case where T = 2.) What is the expected draft position of the team with the Nth-best record?

Here is my solution to the general case:
[Show Solution]

While the expected draft position is not that difficult to characterize, one can also ask about the distribution of draft positions:
[Show Solution]

Splitting a hundred dollar bill

This Riddler puzzle investigates a method for deciding who should get a $100 bill found on the ground. It leads to some interesting consequences…

You and four statistician colleagues find a \$100 bill on the floor of your department’s faculty lounge. None of you have change, so you agree to play a game of chance to divide the money probabilistically. The five of you sit around a table. The game is played in turns. Each turn, one of three things can happen, each with an equal probability: The bill can move one position to the left, one position to the right, or the game ends and the person with the bill in front of him or her wins the game. You have tenure and seniority, so the bill starts in front of you. What are the chances you win the money? What if there were N statisticians in the department?

Here is my solution to the first part, assuming five statisticians.
[Show Solution]

Here is my solution to the second part, assuming $N$ statisticians.
[Show Solution]

For the brave and curious, this next section explores connections between the problem and Fourier Transforms, complex analysis, and Chebyshev polynomials. Fair warning: advanced math!
[Show Solution]

What if robots cut your pizza?

This Riddler puzzle is about random chords of a circle and the regions they describe.

At RoboPizza™, pies are cut by robots. When making each cut, a robot will randomly (and independently) pick two points on a pizza’s circumference, and then cut along the chord connecting them. If you order a pizza and specify that you want the robot to make exactly three cuts, what is the expected number of pieces your pie will have?

Here is a simple solution, which was pointed out to me in a comment to my original post.
[Show Solution]

The following solution is a bit more complicated, and computes the entire distribution rather than just its expected value.
[Show Solution]

If you’ve already read the solution above and you’re interested in the distribution of pieces for the general case, read on!
[Show Solution]

Monsters’ gems

Once again, The Riddler does not disappoint! This puzzle is about slaying monsters and collecting gems.

A video game requires you to slay monsters to collect gems. Every time you slay a monster, it drops one of three types of gems: a common gem, an uncommon gem or a rare gem. The probabilities of these gems being dropped are in the ratio of 3:2:1 — three common gems for every two uncommon gems for every one rare gem, on average. If you slay monsters until you have at least one of each of the three types of gems, how many of the most common gems will you end up with, on average?

Here is my solution:
[Show Solution]

A more brute-force approach:
[Show Solution]

Yet another solution approach with very nice write-up can be found on Andrew Mascioli’s blog

Elevator button puzzle

This problem was originally posted on the Riddler blog. Here it goes:

In a building’s lobby, some number (N) of people get on an elevator that goes to some number (M) of floors. There may be more people than floors, or more floors than people. Each person is equally likely to choose any floor, independently of one another. When a floor button is pushed, it will light up.

What is the expected number of lit buttons when the elevator begins its ascent?

My solution:
[Show Solution]

A much more elegant solution, courtesy of Ross Boczar
[Show Solution]