https://thirtysixrolls.com/ Probability you can feel: interactive explanations of dice, evidence, bots and chance. en Sat, 03 Oct 2026 14:49:02 +0000 https://thirtysixrolls.com/#/a/bots-tree https://thirtysixrolls.com/#/a/bots-tree Bots Chess programs look far ahead. Backgammon programs mostly can't, and that changed how they had to think. Sat, 03 Oct 2026 14:49:02 +0000 https://thirtysixrolls.com/#/a/bots-handtuned https://thirtysixrolls.com/#/a/bots-handtuned Bots In 1979 a program on a Carnegie-Mellon mainframe beat Luigi Villa 7–1. Its knowledge was written by people. Sat, 03 Oct 2026 14:49:02 +0000 https://thirtysixrolls.com/#/a/bots-encoding https://thirtysixrolls.com/#/a/bots-encoding Bots Before a network can judge a position, the position has to become a list of numbers. TD-Gammon used 198. Sat, 03 Oct 2026 14:49:02 +0000 https://thirtysixrolls.com/#/a/bots-network https://thirtysixrolls.com/#/a/bots-network Bots 198 numbers in, five probabilities out. What happens in between, for any position you like. Sat, 03 Oct 2026 14:49:02 +0000 https://thirtysixrolls.com/#/a/bots-td https://thirtysixrolls.com/#/a/bots-td Bots No expert games, no rules of thumb. Just a network that keeps comparing what it predicted with what it predicts next. Sat, 03 Oct 2026 14:49:02 +0000 https://thirtysixrolls.com/#/a/bots-moves https://thirtysixrolls.com/#/a/bots-moves Bots Generate every legal play, judge each result, then look one roll deeper for the close calls. Sat, 03 Oct 2026 14:49:02 +0000 https://thirtysixrolls.com/#/a/bots-selfplay https://thirtysixrolls.com/#/a/bots-selfplay Bots Watch the win probability swing on every roll, and see where skill ends and luck begins. Sat, 03 Oct 2026 14:49:02 +0000 https://thirtysixrolls.com/#/a/bots-rollout https://thirtysixrolls.com/#/a/bots-rollout Bots When a snap judgment isn't good enough, play the position out thousands of times and count. Sat, 03 Oct 2026 14:49:02 +0000 https://thirtysixrolls.com/#/a/bots-bearoff https://thirtysixrolls.com/#/a/bots-bearoff Bots Near the end of a race, a computer doesn't estimate. It knows. Sat, 03 Oct 2026 14:49:02 +0000 https://thirtysixrolls.com/#/a/bots-cube https://thirtysixrolls.com/#/a/bots-cube Bots Networks estimate winning chances without a cube. Turning that into cube decisions took a separate theory. Sat, 03 Oct 2026 14:49:02 +0000 https://thirtysixrolls.com/#/a/bots-timeline https://thirtysixrolls.com/#/a/bots-timeline History Forty-odd years of backgammon programs, and the handful of people behind them. Sat, 03 Oct 2026 14:49:02 +0000 https://thirtysixrolls.com/#/a/craps https://thirtysixrolls.com/#/a/craps Craps The loudest table in the casino, the cheapest bet on its felt, why no betting system works, and the dice games you play at home. Sat, 03 Oct 2026 14:49:02 +0000 https://thirtysixrolls.com/#/a/craps-best-bet https://thirtysixrolls.com/#/a/craps-best-bet Craps One bet on the craps table has no house edge at all — but you can only make it by buying in with another bet. Here's what the package really costs. Sat, 03 Oct 2026 14:49:02 +0000 https://thirtysixrolls.com/#/a/craps-systems https://thirtysixrolls.com/#/a/craps-systems Craps Double after every loss and you'll almost always walk away a winner. Almost. Sat, 03 Oct 2026 14:49:02 +0000 https://thirtysixrolls.com/#/a/craps-bold https://thirtysixrolls.com/#/a/craps-bold Craps You have $100 and need $200 by the end of the night. Probability theory has an unexpectedly definite answer. Sat, 03 Oct 2026 14:49:02 +0000 https://thirtysixrolls.com/#/a/craps-hand https://thirtysixrolls.com/#/a/craps-hand Craps In 2009 a grandmother playing craps for the second time held the dice for four hours and eighteen minutes. How unlikely was that? Sat, 03 Oct 2026 14:49:02 +0000 https://thirtysixrolls.com/#/a/yahtzee https://thirtysixrolls.com/#/a/yahtzee Dice games Keep the biggest group, reroll the rest. You'll get a Yahtzee about one turn in 22. Sat, 03 Oct 2026 14:49:02 +0000 https://thirtysixrolls.com/#/a/liars-dice https://thirtysixrolls.com/#/a/liars-dice Dice games Every bid is a claim about dice nobody can see. The binomial distribution tells you when to call. Sat, 03 Oct 2026 14:49:02 +0000 https://thirtysixrolls.com/#/a/chuck-a-luck https://thirtysixrolls.com/#/a/chuck-a-luck Dice games Three dice, pick a number. Three chances at 1 in 6 sounds like even money. It isn't. Sat, 03 Oct 2026 14:49:02 +0000 https://thirtysixrolls.com/#/a/galileo-dice https://thirtysixrolls.com/#/a/galileo-dice Dice games With three dice, 9 and 10 can each be made six ways. So why does 10 come up more often? Sat, 03 Oct 2026 14:49:02 +0000 https://thirtysixrolls.com/#/a/frontiers https://thirtysixrolls.com/#/a/frontiers Frontiers What researchers are actually working on now: recent papers on probability, evidence, machines and the real world, in plain English — with a line of each abstract so you can check us. Sat, 03 Oct 2026 14:49:02 +0000 https://thirtysixrolls.com/#/a/fr-statistics https://thirtysixrolls.com/#/a/fr-statistics Frontiers Prediction intervals that don't assume anything, tests you can watch forever, and fresh evidence that doctors still misread test results. Sat, 03 Oct 2026 14:49:02 +0000 https://thirtysixrolls.com/#/a/fr-ai https://thirtysixrolls.com/#/a/fr-ai Frontiers Search with hidden information, poker bots that learn to be deliberately random, and a warning about believing what a model says its risk is. Sat, 03 Oct 2026 14:49:02 +0000 https://thirtysixrolls.com/#/a/fr-applied https://thirtysixrolls.com/#/a/fr-applied Frontiers A fraud test that flags honest data, a forecasting victory that evaporates, and the first causal evidence that the friendship paradox changes behaviour. Sat, 03 Oct 2026 14:49:02 +0000 https://thirtysixrolls.com/#/a/fr-method https://thirtysixrolls.com/#/a/fr-method Frontiers What we checked, what we didn't, and why most of these are preprints. Sat, 03 Oct 2026 14:49:02 +0000 https://thirtysixrolls.com/#/a/gammonster https://thirtysixrolls.com/#/a/gammonster Lab notebook A lab notebook: our own attempt at a strong backgammon bot — the bugs that wasted months, the benchmarks that found them, and a network you can play. Sat, 03 Oct 2026 14:49:02 +0000 https://thirtysixrolls.com/#/a/gm-bugs https://thirtysixrolls.com/#/a/gm-bugs Lab notebook The search never searched, the game started from the wrong position, and the rules couldn't hit. Sat, 03 Oct 2026 14:49:02 +0000 https://thirtysixrolls.com/#/a/gm-measure https://thirtysixrolls.com/#/a/gm-measure Lab notebook A falling training loss proves your model fits its data. Only a match proves it plays. Sat, 03 Oct 2026 14:49:02 +0000 https://thirtysixrolls.com/#/a/gm-rebuild https://thirtysixrolls.com/#/a/gm-rebuild Lab notebook A rules engine that agrees with a tested one, then the oldest trick in the book: temporal-difference self-play. Sat, 03 Oct 2026 14:49:02 +0000 https://thirtysixrolls.com/#/a/gm-play-it https://thirtysixrolls.com/#/a/gm-play-it Lab notebook The network runs in your browser and grades every move you make. Sat, 03 Oct 2026 14:49:02 +0000 https://thirtysixrolls.com/#/a/gm-deeper https://thirtysixrolls.com/#/a/gm-deeper Lab notebook Six times faster training, a second roll of look-ahead, wider networks, hand-made features and rollouts — each measured over thousands of games. Sat, 03 Oct 2026 14:49:02 +0000 https://thirtysixrolls.com/#/a/gm-planning https://thirtysixrolls.com/#/a/gm-planning Lab notebook A Stratego result sent us back to a conclusion we'd already written down — and it was wrong. Sat, 03 Oct 2026 14:49:02 +0000 https://thirtysixrolls.com/#/a/chains https://thirtysixrolls.com/#/a/chains Queues Processes with no memory, wanderers who end up where the roads are, and how to measure surprise in bits. Sat, 03 Oct 2026 14:49:02 +0000 https://thirtysixrolls.com/#/a/q-bus https://thirtysixrolls.com/#/a/q-bus Queues Buses every ten minutes should mean a five-minute wait. Unless the gaps vary — and then you're more likely to arrive during a long one. Sat, 03 Oct 2026 14:49:02 +0000 https://thirtysixrolls.com/#/a/q-traffic https://thirtysixrolls.com/#/a/q-traffic Queues At 80% busy the line is fine. At 95% it's long. At 100% it never ends — and the change is not gradual. Sat, 03 Oct 2026 14:49:02 +0000 https://thirtysixrolls.com/#/a/c-markov https://thirtysixrolls.com/#/a/c-markov Chains Tomorrow depends on today and nothing before it. That one rule describes weather, board games, and why shuffling works. Sat, 03 Oct 2026 14:49:02 +0000 https://thirtysixrolls.com/#/a/c-walk https://thirtysixrolls.com/#/a/c-walk Chains Wander a map at random and you'll spend your time in proportion to how connected each place is. Google was built on this. Sat, 03 Oct 2026 14:49:02 +0000 https://thirtysixrolls.com/#/a/c-bits https://thirtysixrolls.com/#/a/c-bits Information A yes/no answer carries at most one bit. Guessing a number from 1 to 100 needs about 6.64 of them. Sat, 03 Oct 2026 14:49:02 +0000 https://thirtysixrolls.com/#/a/c-entropy https://thirtysixrolls.com/#/a/c-entropy Information Claude Shannon's 1948 answer: rarer outcomes carry more information, and a source's entropy is its average surprise. Sat, 03 Oct 2026 14:49:02 +0000 https://thirtysixrolls.com/#/a/x-floods https://thirtysixrolls.com/#/a/x-floods Extremes It doesn't mean once a century. Over a 30-year mortgage, the chance of at least one is about 26%. Sat, 03 Oct 2026 14:49:02 +0000 https://thirtysixrolls.com/#/a/qm-chsh https://thirtysixrolls.com/#/a/qm-chsh Randomness A game two players can win 75% of the time by planning ahead — and about 85% of the time with entangled particles. The gap is how you prove randomness is real. Sat, 03 Oct 2026 14:49:02 +0000 https://thirtysixrolls.com/#/a/positive-test https://thirtysixrolls.com/#/a/positive-test Evidence A test that catches 90% of cases, with few false alarms, can still be wrong most of the time it says yes. Sat, 03 Oct 2026 14:49:02 +0000 https://thirtysixrolls.com/#/a/regression-to-mean https://thirtysixrolls.com/#/a/regression-to-mean Evidence The top scorers on one test usually do worse on the next. Nobody got lazier. Sat, 03 Oct 2026 14:49:02 +0000 https://thirtysixrolls.com/#/a/jelly-beans https://thirtysixrolls.com/#/a/jelly-beans Evidence Run enough studies on nothing and some will look like discoveries. Sat, 03 Oct 2026 14:49:02 +0000 https://thirtysixrolls.com/#/a/confidence-intervals https://thirtysixrolls.com/#/a/confidence-intervals Evidence Not that there's a 95% chance the truth is in this particular range — something subtler, and easier to see. Sat, 03 Oct 2026 14:49:02 +0000 https://thirtysixrolls.com/#/a/simpsons-paradox https://thirtysixrolls.com/#/a/simpsons-paradox Evidence How one treatment can win for small kidney stones and for large ones, yet lose when you add them up. Sat, 03 Oct 2026 14:49:02 +0000 https://thirtysixrolls.com/#/a/bell-curve https://thirtysixrolls.com/#/a/bell-curve Randomness Add up enough small random nudges and the same shape appears, whatever the nudges were. Sat, 03 Oct 2026 14:49:02 +0000 https://thirtysixrolls.com/#/a/random-clumps https://thirtysixrolls.com/#/a/random-clumps Randomness Truly random dots leave empty spaces and crowded corners. Evenly spread dots are the suspicious ones. Sat, 03 Oct 2026 14:49:02 +0000 https://thirtysixrolls.com/#/a/benfords-law https://thirtysixrolls.com/#/a/benfords-law Patterns In a lot of real data, about 30% of numbers begin with 1 and fewer than 5% with 9. Sat, 03 Oct 2026 14:49:02 +0000 https://thirtysixrolls.com/#/a/buffons-needle https://thirtysixrolls.com/#/a/buffons-needle History A probability puzzle from the 1700s turns a floor full of needles into a way to measure π. Sat, 03 Oct 2026 14:49:02 +0000 https://thirtysixrolls.com/#/a/kelly-criterion https://thirtysixrolls.com/#/a/kelly-criterion Bets Having an edge isn't enough. Bet too much and you go broke anyway. Sat, 03 Oct 2026 14:49:02 +0000 https://thirtysixrolls.com/#/a/gamblers-ruin https://thirtysixrolls.com/#/a/gamblers-ruin Bets A 2.7% edge doesn't sound like much until you place a few thousand bets. Sat, 03 Oct 2026 14:49:02 +0000 https://thirtysixrolls.com/#/a/lottery-odds https://thirtysixrolls.com/#/a/lottery-odds Bets A number too big to feel. Here's a way to feel it. Sat, 03 Oct 2026 14:49:02 +0000 https://thirtysixrolls.com/#/a/optimal-stopping https://thirtysixrolls.com/#/a/optimal-stopping Decisions Hiring, house-hunting, dating: if you can't go back, look at about 37% of the options, then commit to the first that beats them all. Sat, 03 Oct 2026 14:49:02 +0000 https://thirtysixrolls.com/#/a/coupon-collector https://thirtysixrolls.com/#/a/coupon-collector Patterns Why finishing a collection takes so much longer than starting it. Sat, 03 Oct 2026 14:49:02 +0000 https://thirtysixrolls.com/#/a/seven-shuffles https://thirtysixrolls.com/#/a/seven-shuffles Cards Why a deck riffled three or four times is still far from random. Sat, 03 Oct 2026 14:49:02 +0000 https://thirtysixrolls.com/#/a/poker-hands https://thirtysixrolls.com/#/a/poker-hands Cards 2,598,960 possible hands, counted. Sat, 03 Oct 2026 14:49:02 +0000 https://thirtysixrolls.com/#/a/play-the-bot https://thirtysixrolls.com/#/a/play-the-bot Play A full game against the neural network from the Bots course, with a grade for every decision you make. Sat, 03 Oct 2026 14:49:02 +0000 https://thirtysixrolls.com/#/a/history https://thirtysixrolls.com/#/a/history Backgammon A gambler's question in 1654, a self-taught backgammon program in 1992, and the people in between. Sat, 03 Oct 2026 14:49:02 +0000 https://thirtysixrolls.com/#/a/take-point https://thirtysixrolls.com/#/a/take-point Backgammon The doubling cube asks one question: is losing 1 point for sure better than risking 2? Sat, 03 Oct 2026 14:49:02 +0000 https://thirtysixrolls.com/#/a/td-gammon https://thirtysixrolls.com/#/a/td-gammon History In the early 1990s, a neural network at IBM played itself at backgammon and changed how humans open the game. Sat, 03 Oct 2026 14:49:02 +0000 https://thirtysixrolls.com/#/a/sevens https://thirtysixrolls.com/#/a/sevens Basics Roll two dice enough times and a triangle appears out of nowhere. Sat, 03 Oct 2026 14:49:02 +0000 https://thirtysixrolls.com/#/a/birthday https://thirtysixrolls.com/#/a/birthday Paradox In a room of 23 people, it's slightly better than even that two share a birthday. Sat, 03 Oct 2026 14:49:02 +0000 https://thirtysixrolls.com/#/a/monty-hall https://thirtysixrolls.com/#/a/monty-hall Paradox The game-show puzzle that famously fooled a lot of people with math degrees. Sat, 03 Oct 2026 14:49:02 +0000 https://thirtysixrolls.com/#/a/streaks https://thirtysixrolls.com/#/a/streaks Intuition Why a “hot streak” at the table is usually just what fair dice look like. Sat, 03 Oct 2026 14:49:02 +0000 https://thirtysixrolls.com/#/a/efron-dice https://thirtysixrolls.com/#/a/efron-dice Puzzle A beats B, B beats C, C beats D, and D beats A. Always pick second. Sat, 03 Oct 2026 14:49:02 +0000 https://thirtysixrolls.com/#/a/unfinished-game https://thirtysixrolls.com/#/a/unfinished-game History In 1654, a gambler's question to Blaise Pascal started a correspondence with Pierre de Fermat that shaped probability theory. Sat, 03 Oct 2026 14:49:02 +0000 https://thirtysixrolls.com/#/a/friendship-paradox https://thirtysixrolls.com/#/a/friendship-paradox Networks It's not an insult. It's a sampling effect, and it shows up in almost every real network. Sat, 03 Oct 2026 14:49:02 +0000