Pick points at random on a line, a circle or a square, and ask what's likely. Geometric probability turns these into areas and lengths, and the answers are rarely what they look like. Meeting a friend at the station, a triangle that holds the center of a circle, Buffon's needle: draw the picture, then check your gut against the simulation.
What these puzzles practice
Turning probabilities into areas
Symmetry on circles and lines
The inspection paradox in space
13 puzzles, easiest first
Coin on the tiles· Carnival game · difficulty 2 of 5At a carnival booth you toss a coin 1 inch across onto a large floor of square tiles, each 3 inches on a side. The coin lands flat at a…
Meeting at the station· Commuting · difficulty 2 of 5You and a friend agree to meet at the train station between 6:00 and 7:00 pm. Each of you arrives at an independent, uniformly random time…
Two cars, one road· Road trip · difficulty 2 of 5Two cars break down at independent, uniformly random spots along a straight 1-mile country road. What is the expected distance between…
A random quadratic· Algebra class · difficulty 3 of 5Two numbers B and C are chosen independently and uniformly at random between 0 and 1. What is the probability that the equation x² + Bx +…
Across the running track· Athletics · difficulty 3 of 5Two friends stand at independent, uniformly random points on a circular running track of radius R. What is the probability that the…
Break the stick· Geometry · difficulty 3 of 5You break a stick at 2 points chosen independently and uniformly along its length, getting 3 pieces. What is the probability that the 3…
Does it hold the center?· Geometry · difficulty 3 of 5You drop 3 points independently and uniformly at random on a circle, then join them in order around the circle to make a triangle. What is…
How far from the bullseye· Darts · difficulty 3 of 5A point is chosen uniformly at random inside a disc (think of a dartboard) of radius 1. What is its expected distance from the center?
All three at once· Coffee shop · difficulty 4 of 5Three friends each arrive at a coffee shop at an independent, uniformly random time between 1:00 and 2:00 pm. Each stays exactly 20…
Center or edge· Backyard · difficulty 4 of 5A ball lands at a uniformly random point on a square lawn 10 feet on a side. What is the probability that it is closer to the center of…
The needle on the floorboards· Geometry · difficulty 4 of 5A floor is ruled with parallel lines 1 unit apart. You drop a needle of length 1 at a uniformly random position and angle. What is the…
The smooth stretch outside your house· City roads · difficulty 4 of 5A 1,000-foot road has 3 potholes, each at an independent, uniformly random point along it. Your driveway is 500 feet from the start of the…
Two riders, one square· Delivery · difficulty 5 of 5Two delivery riders are at independent, uniformly random spots in a square city sector that measures 1 km on each side. What is the…