Blog posts and worked probability puzzle solutions.https://thealphajas.github.io/enhttps://thealphajas.github.io/fun/probability/party-groups/https://thealphajas.github.io/fun/probability/party-groups/There are 50 guests at a party and they are making groups for a game. To do this they each write their name on a piece of paper and put it into a hat. One by one, each guest picks a name from the hat. Each guest will be a part of a group with the guest they pulled the name out of from the hat. If…Sun, 27 Sep 2026 19:33:45 GMTProbabilityExpectationRecursionRandom PermutationsHarmonic Numbershttps://thealphajas.github.io/fun/probability/take-and-roll-ii/https://thealphajas.github.io/fun/probability/take-and-roll-ii/You are given a fair 20-sided die and 100 actions in a game. The die starts with upface 1. The two options you can perform are to roll and to take. Performing a roll re-rolls the current upface of the die. Performing a take allows you to cash out the current upface of the die. Note that the game…Sun, 27 Sep 2026 19:05:25 GMTProbabilityExpectationMarkov ChainsGeometric DistributionOptimizationhttps://thealphajas.github.io/fun/probability/decreasing-uniform-chain/https://thealphajas.github.io/fun/probability/decreasing-uniform-chain/Let X1, X2, … ~ Unif(0,1) IID. Let N be the first index n where Xn ≠ min {X1,…,Xn}. Find E[X_(N-1)], i.e. the smallest value among the first N-1 values selected. The answer will be in the form a + be for integers a and b, where e is Euler's constant. Find a+b.Fri, 25 Sep 2026 09:55:03 GMTProbabilityOrder StatisticsRecord ValuesExpectationhttps://thealphajas.github.io/fun/probability/power-grid/https://thealphajas.github.io/fun/probability/power-grid/A 3 × 3 grid of light bulbs is formed. Then, each light bulb is powered on with probability 1/2. Find the probability that no two adjacent (grid cells that share a common side) light bulbs are powered on.Fri, 25 Sep 2026 09:55:03 GMTProbabilityDynamic ProgrammingCombinatoricsTransfer Matrix Methodhttps://thealphajas.github.io/fun/probability/dice-coin-paradigm/https://thealphajas.github.io/fun/probability/dice-coin-paradigm/We flip a fair coin until we obtain our first tails. Given the first tails occurs on the nth flip, we roll a fair 6-sided die n times. Find the probability that the die value 1 is observed in the rolls.Mon, 14 Sep 2026 15:30:39 GMTProbabilityConditional ProbabilityGeometric SeriesInfinite Serieshttps://thealphajas.github.io/fun/probability/local-maxima/https://thealphajas.github.io/fun/probability/local-maxima/14 pieces of paper labelled 1–14 are placed in a line at random. We spot i is a local maxima if the paper at the ith position is strictly larger than all of its adjacent papers. Find the expected number of local maxima in the sequence. For example, with 6 numbers, 513246 has 3 local maxima, at the…Mon, 14 Sep 2026 15:30:39 GMTProbabilityLinearity of ExpectationSymmetryRecursionhttps://thealphajas.github.io/fun/probability/overlapping-subsets/https://thealphajas.github.io/fun/probability/overlapping-subsets/Consider the set Ω = {1, 2, …, 20}. Two subsets A and B of Ω are uniformly at random selected from the power set of Ω, independently. It is possible that A = B. Define N = |A ∩ B|. Compute Var(N)/E[N].Mon, 14 Sep 2026 15:30:39 GMTProbabilityBinomial DistributionGenerating FunctionsVariancehttps://thealphajas.github.io/fun/probability/contracts-and-pricing-iii/https://thealphajas.github.io/fun/probability/contracts-and-pricing-iii/After being caught in an embarrassing high-profile data fabrication scandal, Calambya University decides to stop sharing data with the national college ranking organization, effective in 2023. Rival school Mahogany University is overjoyed and offers each student a tuition refund worth X thousand…Fri, 11 Sep 2026 10:56:20 GMTProbabilityExpectationExponential DistributionOptions Pricinghttps://thealphajas.github.io/fun/blogs/monotonic-search-with-non-uniform-priors/https://thealphajas.github.io/fun/blogs/monotonic-search-with-non-uniform-priors/Binary search always queries the midpoint — but that's only optimal when you're clueless about where the answer lies. Here's the information-theoretic argument for querying the median of your belief instead.Sun, 06 Sep 2026 00:00:00 GMTBlogInformation TheorySearch AlgorithmsBayesian Statisticshttps://thealphajas.github.io/fun/blogs/why-binary-has-to-be-the-way-it-is/https://thealphajas.github.io/fun/blogs/why-binary-has-to-be-the-way-it-is/Put three natural constraints on a place-value representation system and the powers of two are not a choice any more — they are forced.Tue, 18 Aug 2026 00:00:00 GMTBlogNumber SystemsCombinatoricsProofshttps://thealphajas.github.io/fun/probability/basketball-practice-ii/https://thealphajas.github.io/fun/probability/basketball-practice-ii/Frank makes his first free throw and misses his second free throw. The probability of making the nth free throw is equal to the proportion of free throws he made during his first n-1 attempts. How many free throws can Frank expect to make in 100 attempts?Wed, 12 Aug 2026 17:03:13 GMTProbabilityExpectationConditional Expectationhttps://thealphajas.github.io/fun/probability/consecutive-tails/https://thealphajas.github.io/fun/probability/consecutive-tails/A fair coin is flipped 10 times. What is the probability that any tails which occur, occur only in consecutive pairs? For instance, THHTTHHTTH fails because of the lone tail at the start, while HHTTTTHHHH is fine — as is all heads.Wed, 12 Aug 2026 17:03:13 GMTProbabilityCombinatoricshttps://thealphajas.github.io/fun/probability/place-or-take/https://thealphajas.github.io/fun/probability/place-or-take/You are playing a one-player game with two opaque boxes. At each turn, you can choose to either "place" or "take". "Place" places $1 from a third party into one box randomly. "Take" empties out one box randomly and that money is yours. This game consists of 100 turns where you must either place or…Wed, 12 Aug 2026 17:03:13 GMTProbabilityExpectationSeriesOptimizationhttps://thealphajas.github.io/fun/probability/show-the-face/https://thealphajas.github.io/fun/probability/show-the-face/Roll a fair standard 6-sided die until a 6 appears. Given that the first 6 occurs before the first 5, find the expected number of times the die was rolled.Wed, 12 Aug 2026 17:03:13 GMTProbabilityExpectationConditional ProbabilityMarkov ChainsGeometric Distributionhttps://thealphajas.github.io/fun/probability/breaking-stick/https://thealphajas.github.io/fun/probability/breaking-stick/A stick of length L drops and breaks at a random point distributed uniformly across the length. What is the expected length of the smaller part?Mon, 08 Dec 2025 11:44:19 GMTProbabilityExpectationUniform DistributionIntegrationhttps://thealphajas.github.io/fun/probability/chess-tournament-i/https://thealphajas.github.io/fun/probability/chess-tournament-i/A chess tournament has 128 players, each with a distinct rating. Assume that the player with the higher rating always wins against a lower rated opponent and that the winner proceeds to the subsequent round. What is the probability that the highest rated and second-highest rated players will meet…Mon, 08 Dec 2025 11:44:19 GMTProbabilityCombinatoricshttps://thealphajas.github.io/fun/probability/comparing-flips-ii/https://thealphajas.github.io/fun/probability/comparing-flips-ii/You and your friend are playing a game with a fair coin, tossing it and writing down the outcomes. You win if HTH appears before HHT, else your friend wins. What is the probability that your friend wins?Mon, 08 Dec 2025 11:44:19 GMTProbabilityConditional ProbabilityMarkov Chainshttps://thealphajas.github.io/fun/probability/confused-ant-ii/https://thealphajas.github.io/fun/probability/confused-ant-ii/An ant walks the corner of a 3D cube and moves to one of the three adjacent vertices with equal probability at each step. Find the expected number of steps needed for the ant to return to the vertex it started at.Mon, 08 Dec 2025 11:44:19 GMTProbabilityExpectationMarkov ChainsRandom Walkshttps://thealphajas.github.io/fun/probability/graph-search-i/https://thealphajas.github.io/fun/probability/graph-search-i/You are given an undirected graph with 10 nodes. From every node, you are able to access any other node (including itself), all with an equal probability of 1/10. What is the expected number of steps to reach all nodes at least once (rounded to the nearest step)?Mon, 08 Dec 2025 11:44:19 GMTProbabilityExpectationCoupon Collectorhttps://thealphajas.github.io/fun/probability/graph-search-ii/https://thealphajas.github.io/fun/probability/graph-search-ii/You are given an undirected graph with 11 nodes. From every node, you are able to access any other node (not including itself), all with an equal probability of 1/10. What is the expected number of steps to reach all nodes at least once (rounded to the nearest step)?Mon, 08 Dec 2025 11:44:19 GMTProbabilityExpectationCoupon Collectorhttps://thealphajas.github.io/fun/probability/intersecting-chords/https://thealphajas.github.io/fun/probability/intersecting-chords/10 chords with uniformly randomly chosen endpoints are drawn on a circle. What is the expected number of intersections?Mon, 08 Dec 2025 11:44:19 GMTProbabilityExpectationCombinatoricshttps://thealphajas.github.io/fun/probability/matching-die-trio/https://thealphajas.github.io/fun/probability/matching-die-trio/Three fair 6−sided dice are rolled and their upfaces are recorded. Find the probability that the values showing upon rolling all three dice again is the same as the original three values recorded.Mon, 08 Dec 2025 11:44:19 GMTProbabilityCombinatoricshttps://thealphajas.github.io/fun/probability/mean-babysitter/https://thealphajas.github.io/fun/probability/mean-babysitter/10 kids are really hungry! Their babysitter has 12 units of food to give. However, she decides she only wants to give 4 of the children food. How many ways can she distribute the food units such that 6 of the children are hungry (receive no food), and the other 4 children receive at least 1 unit…Mon, 08 Dec 2025 11:44:19 GMTProbabilityCombinatoricshttps://thealphajas.github.io/fun/probability/modified-rng/https://thealphajas.github.io/fun/probability/modified-rng/Jimmy picks a number uniformly at random from (0,1). If Jimmy chooses x, then Jon picks a number from (x,1) uniformly at random. If Y represents the number Jon selects, in simplest form, find E[Y]/Var(Y)Mon, 08 Dec 2025 11:44:19 GMTProbabilityConditional ExpectationUniform Distributionhttps://thealphajas.github.io/fun/probability/multinomial-expansion/https://thealphajas.github.io/fun/probability/multinomial-expansion/How many terms are there in the expansion of (x1 + x2 + x3 + x4 + x5 + x6)^18 after all like terms have been combined?Mon, 08 Dec 2025 11:44:19 GMTProbabilityCombinatoricshttps://thealphajas.github.io/fun/probability/multinomial-sum/https://thealphajas.github.io/fun/probability/multinomial-sum/Find the sum of all multinomial coefficients: C(7; b1, b2, b3, b4) where b1 + b2 + b3 + b4 = 7 and each bi ≥ 0 is an integer.Mon, 08 Dec 2025 11:44:19 GMTProbabilityCombinatoricshttps://thealphajas.github.io/fun/probability/optimal-bidders-i/https://thealphajas.github.io/fun/probability/optimal-bidders-i/Carter has come into contact with a bounty of gold. He takes it to an auction shop. The shop tells him that every bidder will place a bid uniformly between 500 and1000. They also say they can recruit bidders for $5 per person. Carter needs to figure out the optimal number of bidders to maximize…Mon, 08 Dec 2025 11:44:19 GMTProbabilityExpectationOrder StatisticsOptimizationhttps://thealphajas.github.io/fun/probability/particle-reach/https://thealphajas.github.io/fun/probability/particle-reach/Consider a particle that performs a random walk on the integers starting at position 0. At each step, the particle moves from position i to position i+1 with probability p, while the probability it moves from i to i−1 is 1−p. If p=1/3, find the probability the particle ever reaches position 1Mon, 08 Dec 2025 11:44:19 GMTProbabilityRandom WalksConditional Probabilityhttps://thealphajas.github.io/fun/probability/pass-the-ball/https://thealphajas.github.io/fun/probability/pass-the-ball/You and 4 other people are sitting in a circle. You are given a ball to start the game. Every second of this game, the person with the ball has three choices they can make. They can either pass the ball to the left, pass the ball to the right, or keep the ball (all with equal probability). This…Mon, 08 Dec 2025 11:44:19 GMTProbabilityConditional ProbabilityMarkov Chainshttps://thealphajas.github.io/fun/probability/random-angle-ii/https://thealphajas.github.io/fun/probability/random-angle-ii/A right triangle is being formed with legs labeled A and B. The random lengths of legs A and B are both IID Uniform(0,1) random variables. Let θ be the angle (in radians) opposite to side A. Find the probability that: P(θ > π /3)Mon, 08 Dec 2025 11:44:19 GMTProbabilityUniform DistributionIntegrationGeometric Probabilityhttps://thealphajas.github.io/fun/probability/random-ratio/https://thealphajas.github.io/fun/probability/random-ratio/p and q are two points chosen at random between 0 and 1. What is the probability that the ratio p/q lies between 1 and 2?Mon, 08 Dec 2025 11:44:19 GMTProbabilityUniform DistributionIntegrationGeometric Probabilityhttps://thealphajas.github.io/fun/probability/russian-roulette-i/https://thealphajas.github.io/fun/probability/russian-roulette-i/You're playing a game of Russian Roulette with a friend. The six-chambered revolver is loaded with one bullet. Initially, the cylinder is spun to randomize the order of the chambers. The two of you take turns pulling the trigger until the person who fires the gun loses. Given that the cylinder is…Mon, 08 Dec 2025 11:44:19 GMTProbabilityConditional Probabilityhttps://thealphajas.github.io/fun/probability/russian-roulette-ii/https://thealphajas.github.io/fun/probability/russian-roulette-ii/You're playing a game of Russian Roulette with a friend. The six-chambered revolver is loaded with one bullet. Initially, the cylinder is spun to randomize the order of the chambers. The two of you take turns pulling the trigger until the person who fires the gun loses. Given that the cylinder is…Mon, 08 Dec 2025 11:44:19 GMTProbabilitySeriesConditional Probabilityhttps://thealphajas.github.io/fun/probability/square-cross/https://thealphajas.github.io/fun/probability/square-cross/A square of side length 20 is placed in front of you. You select a point uniformly at random from its interior. Then, independently, a circle of radius R ~ Unif(0,10) is formed around the selected point. What is the probability that this circle does not intersect the square at any point?Mon, 08 Dec 2025 11:44:19 GMTProbabilityUniform DistributionIntegrationGeometric Probabilityhttps://thealphajas.github.io/fun/probability/differ-by-2/https://thealphajas.github.io/fun/probability/differ-by-2/How many times do we have to roll a fair 6-sided die till we roll two numbers in a row that differ by 2?Mon, 08 Dec 2025 11:44:19 GMTProbabilityExpectationMarkov Chainshttps://thealphajas.github.io/fun/probability/exact-5-ii/https://thealphajas.github.io/fun/probability/exact-5-ii/Abd continually rolls a fair 6-sided die until he obtains his first 6. What’s the expected number of times Abd rolls a 5 before he stops?Mon, 08 Dec 2025 11:44:19 GMTProbabilityExpectationGeometric Distribution