Geometric probability
Coin in square
The following game is often offered at carnivals. There is a table whose surface is ruled in 1-inch squares. You take a penny from your pocket (3/4 inch diameter), and after stepping several feet away from the table, you throw the penny on top of it. If the penny falls within a single square, you get 5 cents and your penny back. If the penny falls on one of the rulings, you loose your penny. Would you play this game?
Serendipity
After talking about the iconic movies "Serendipity" and "Sleepless in Seattle" on a dating website, a potential couple decides to go on a random date. Each party will show up at a predetermined restaurant randomly between 7 and 8 pm, and wait for exactly 15 minutes, but no later than 8 pm. If they run into each other, the relationship was meant to be, and otherwise, they move on. What is the chance of success?
Random chord
If we select a chord randomly on a fixed circle, what is the probability that its length will exceed the radius of the circle?
Points in a semicircle
If we choose points randomly on a fixed circle, what is the probability they will lie in a semicircle?
Landing on the side
How thick should a coin be so that it lands with probability 1/3 on its side?
Upright table
A carpenter with a good sense of humor attached three legs to a circular table-top in random uniformly distributed positions. What is the chance the table will stay upright?
Estimating
Find a way to estimate $\pi$ using an apparatus which generates random uniformly distributed numbers in the unit interval.
Point on a segment
The unit interval is cut in two using a randomly chosen point. What is the expected length of the shorter piece? What about the longer one?
Two points on a segment
The unit interval is cut at two two independently chosen random points. What is the expected length of the left piece?
Simplex
Suppose that are independent random variables with the uniform distribution on . What is the probability that ?