This is a compilation of some of my favorite problems from probability and statistics. Some can be done with a bit of intuition, but most require math. When you’re ready, click on the title for an answer and explanation. (If you liked these, check out my logic puzzles!)

### Contents

### Easy

### Medium

### Hard

### Easy

- St. Petersburg Lottery
This casino game involves flips of a fair coin. The pot starts with 2 cents. The player will continue flipping the coin until it lands tails. Every time the coin lands heads, the pot doubles; otherwise, the player keeps the money in the pot and the game ends. Altogether, the player wins 2k dollars, where k is the number of tosses. What’s a fair price for entering the game?

- Two Envelopes
You are given two envelopes with money; one contains twice as much as the other. You may pick one envelope and keep the money it contains. Having chosen an envelope, you are given the chance to switch. Your friend argues that it is always beneficial to switch: “the other envelope contains 2A with probability 1/2 and A/2 with probability 1/2, so the expected value of the other envelope is: (1/2)(2A) + (1/2)(A/2) = (5/4)A.” What is wrong with this argument?

- Coupon Collector
Given n coupons, how many coupons do you expect to draw with replacement before having drawn each coupon at least once? (needs math)

### Medium

- Bus Wait Time
Buses arrive at the bus stop every 15 minutes on the dot. You arrive to the stop at a random time (uniformly distributed). How long should you expect to wait until the next bus shows up? What if the time is Poisson-distributed with a mean wait time of 15 minutes?

- Amoebas Revisited
You place a single amoeba into a jar with infinite space and nutrients. After 3 minutes, that amoeba will either die, persist, or divide with equal probability. In other words, it is equally likely to be replaced by 0, 1, or 2 amoeba in the next generation. What is the probability that the entire population eventually dies out? Bonus: what if it can also divide into 3 offspring? (needs math)

- Ants on a Stick
There are

*N*ants on a meterstick. The position of each ant is uniformly distributed along the stick. Each ant travels at constant speed and takes 1 minute to walk the length of the stick. At time t=0 min, each ant randomly chooses a direction (left or right are equally likely) and begins walking in that direction until it either falls off the stick or bumps into another ant. If it bumps into an ant, both switch directions.

Part 1: In terms of*N*, what is the maximum time it could take for all the ants to fall off?

Part 2: What is the average amount of time it takes for a random initial configuration of N ants to all fall off the stick? (needs math)

### Hard

- 100 Prisoners
The prison warden offers 100 death row prisoners, who are numbered from 1 to 100, a last chance. A room contains a cupboard with 100 drawers. The warden randomly puts one prisoner’s number in each closed drawer. The prisoners enter the room, one after another. Each prisoner may open and look into 50/100 drawers, in any order. The drawers are closed again afterwards. If, during this search, every prisoner finds his number, all prisoners are pardoned. Otherwise, all prisoners die. The prisoners may discuss strategy before the game begins. What is the prisoners’ best strategy? (needs a lot of math to determine the actual probability, but not to find the strategy)