All Problems

Adding the weekday of a boy's birth seems irrelevant. It moves the answer most of the way from 1/3 to 1/2.

An exponential timer races a uniform one. The expected time until the first one goes off is a famous constant.

Thirteen cards, four suits. How often is one suit missing entirely?

Five random numbers in [0, 1]. How often does their median land in the top quarter?

Keep a running total of die rolls and stop the first time it is divisible by 7. How many rolls should you expect?

Multiply two random numbers from [0, 1]. How often is the product below one half?

Roll a fair die until some face shows up for the second time. How many rolls should you expect?

A random point on the unit sphere. How often does the sum of its three coordinates exceed 1?

Three random points on a circle of radius 1 make a triangle. How big is it on average?

Three lengths drawn uniformly from [0, 1]. How often are they the sides of an acute triangle?

Three points are chosen uniformly at random on a circle. What is the probability all three lie within some semicircle?

Each minute an amoeba dies, stays the same, splits in two, or splits in three, each with probability 1/4. What is the probability the population dies out?

Break a stick at two random points. How likely are the pieces to form not just a triangle, but an acute one?

Pick a random point in a square and look at the bottom edge. When does that edge fill more than a right angle of your view?

An ant walks randomly along the edges of a cube. How many moves does it expect to need to reach the opposite vertex?

Two random points in a unit cube. What is the expected distance between them, to four decimal places?

Two random points inside a disk of radius 1. What is the expected distance between them?

Two random points in a unit square. What is the expected distance between them, to four decimal places?

A random chord is drawn by picking two points on a circle. What is the probability it is longer than a side of the inscribed equilateral triangle?

Draw three independent standard normals and keep the largest. What is its expected value, exactly?

Two standard normal scores with correlation one half. How often are both positive?

Snap a stick at random, then snap the longer part at random. How likely are the three pieces to form a triangle?

Drop a needle onto a floor ruled with lines exactly one needle-length apart. What is the probability it crosses a line?

A random point in a square: is it closer to the centre or to the nearest edge? The answer is far from even.

A mouse wanders a 3 × 3 grid of rooms. Starting in a corner, how many moves until it finds the cheese in the middle?

Bertrand's paradox gets a fourth contender: a random point and a random direction. How often is the chord long?

A coin is tossed onto a tiled floor. How likely is it to land clear of every grout line?

Take two independent standard normals and sort them. How correlated are the larger and the smaller?

There are 6 distinct coupons, each equally likely in a cereal box. How many boxes do you expect to buy to collect all 6?

Four random points on a circle, joined in pairs by two chords. How often do the chords cross?

A random chord slices a round pizza in two. What fraction of the pizza is the smaller slice, on average?

What is the probability of rolling at least one six in four rolls of a fair die, the bet that helped launch probability theory?

A fair die is rolled six times. How many distinct faces do you expect to see?

Shuffle the numbers 1 through 10 uniformly. How many pairs end up in the wrong relative order, on average?

Three rods whose lengths are independent exponentials. How likely are they to make a triangle?

A fair die is rolled once and you are paid the square of the face in dollars. What entry fee makes the game fair?

Three machines with different average lifetimes. What is the chance the most durable one fails first and the most fragile one fails last?

Alice and Bob alternate flipping a fair coin, Alice first; whoever flips heads first wins. What is the probability Alice wins?

Four random points in a disk. How likely is it that one of them hides inside the triangle of the other three?

What is the probability a random 5-card poker hand is a full house?

A gambler starts with $10 and bets $1 on a fair coin flip each round, stopping at $0 or $20. What is the probability of reaching $20?

Three boxes, six coins, one gold coin in your hand. The obvious answer about its neighbour is wrong.

Your bridge partner admits to holding the ace of spades. How likely is it that they hold another ace as well?

A fair-coin gambler with $3 plays until broke or at $10. How many rounds does the session last on average?

A dart's horizontal and vertical errors are independent standard normals. How often does it land within one unit of the bullseye?

Ten red and ten blue balls are lined up in random order. How many neighbouring pairs share a colour, on average?

Roll two fair dice. What is the expected value of the larger of the two values?

Colour each of the 15 edges joining six points red or blue by a fair coin. How many single-coloured triangles appear on average?

You pick 1 of 4 doors. The host opens 2 losing doors. What is the probability of winning if you switch?

A random point in an equilateral triangle of height 1. How far is it, on average, from the closest side?

Buffon's needle, but on a floor of square tiles. How often does a tile-length needle touch a grout line?

A fair coin is flipped 10 times. What is the probability no two heads land consecutively?

Three random points in a unit square. How likely is the triangle they form to be obtuse?

A full deck is dealt to four bridge players. How likely is it that the four aces are shared out evenly?

A couple keeps having children until they have at least one boy and at least one girl. How many children do they expect to have?

Mark one random point on each side of a triangle and join them up. How much of the original area does the inner triangle take, on average?

Roll a die up to three times; after each roll, keep the value or roll again. What is the expected payout under optimal play?

Roll a die for dollars, but you may pay a dollar to reroll as often as you like. What is the game worth with best play?

Five friends pass a ball, each time to a random one of the other four. How long until a particular friend gets it?

Line up the numbers 1 to 10 at random. How many of them stand taller than every neighbour, on average?

Draw a ball, return it with another of the same color, repeat. What is the probability the third draw is red?

A symmetric random walk on the integers starts at 0. What is the expected number of steps to return to 0?

Pick the three coefficients of a quadratic at random from [0, 1]. How likely are its roots to be real?

A hand on a clock face jumps one hour forward or back at random. Starting at 12, how long until it is back at 12?

Roll a fair die until a six appears, then stop. What is the expected sum of all your rolls?

Divide one random number by another and round to the nearest integer. How often is the result even?

Two dice are rolled repeatedly. What is the probability a sum of 7 shows up before a sum of 8?

Twenty-three strangers probably contain a shared birthday. How many do you need to probably find one sharing yours?

The chance of seeing a shooting star in an hour is 0.84. What is the chance of seeing one in 10 minutes?

Throw six dice together, again and again, until all six show different faces. How many throws should you expect?

Two shuffled decks are turned over in step. It is rare to get through all 52 without a rank match, but how rare?

Three random points on a unit segment. How likely is every pair to be at least a quarter apart?

Three random numbers in [0, 1]. How often is the square of the third larger than the product of the first two?

The St. Petersburg game promises an infinite expectation, but a real casino can only pay what it has. What is the game actually worth?

Tomorrow's weather depends only on today's. Over many years, what fraction of days are sunny?

A careless secretary stuffs ten letters into ten addressed envelopes at random. How many letters reach the right person, on average?

Four random points on a sphere form a tetrahedron. How likely is it to contain the sphere's centre?

Three random points split a circle into three arcs. A fixed point sits on one of them. Is that arc a typical third?

Two random points on a circle of radius 1. How far apart are they, on average, as the crow flies?

A stick is broken at two uniformly random points. What's the probability the three pieces form a triangle?

Two players each roll a die, and the lower roller pays the higher roller the difference in dollars. How much money changes hands on average?

A gambler with an edge starts with $3 and plays $1 bets until broke or at $6. How likely is success?

Two identical components start together. How long, on average, does the survivor outlive the first failure?

Three points drawn from a standard bivariate normal distribution. How likely is their triangle to be obtuse?

Three random points on a circle form a triangle. How large, on average, is its biggest angle?

Snap a stick at three random points. How long is the longest of the four pieces, on average?

100 passengers, 100 assigned seats, and one lost ticket. What is the probability the last passenger ends up in their own seat?

Pick a random point in a disk and draw the chord that has it as its midpoint. How long is that chord on average?

Stars are scattered at random, one per unit area on average. How far away is the nearest one?

Cars park at random along an endless kerb until no gap fits another car. What fraction of the kerb ends up covered?

Five cards in a row, but not all of one suit. How often does a random poker hand deliver?

Calls arrive at a help desk as a Poisson process averaging six per hour. What is the chance of ten silent minutes?

Divide one standard normal by another. How often is the ratio below the square root of three?

Two friends each arrive at a random time between noon and 1 pm and wait 15 minutes. What is the probability they meet?

You interview n candidates one by one and must decide immediately after each interview. What is the probability the optimal stopping rule selects the best candidate?

Two socks drawn at random from a drawer of red and black socks are both red with probability exactly 1/2. What is the smallest possible drawer?

The warden names a fellow prisoner who will hang. Whether that news should cheer you depends on the warden's habits.

A walker steps forward with probability 2/3 and back with probability 1/3. How long, on average, to get ten steps ahead?

A bulb has already outlived its average lifetime. How long should you expect it to have lasted in total?

You arrive at a bank with two tellers, both busy. What are the odds you are the last of the three customers to leave?

Flip a fair coin until the last three flips agree, all heads or all tails. How many flips do you expect?

Three friends drift into a cafe at random times within an hour and each stays a quarter of an hour. Are all three ever there together?

Twenty-three people suffice for a shared birthday. How many until three people share one?

Three random points in a unit square. What is the expected area of their triangle, as an exact fraction?

Two identical batteries are switched on together. How likely is one to last more than twice as long as the other?

Three bus lines stop at your corner as independent Poisson processes. What is the chance the next two buses are both the rarest line?

A family has two children and you learn at least one is a boy. What is the probability both are boys?

Two marks land at random on a one-metre ruler. How far apart should you expect them to be?

Two random points on a unit sphere. What is the average straight-line distance through the sphere between them?

Flip a fair coin repeatedly. How many flips do you expect to need before seeing two heads in a row?

Flip a fair coin until heads, tails, heads appears in consecutive flips. How long is the wait on average?