{"meta":{"copyright":"© Odds The Law (oddsthelaw.com). Precomputed dataset; all rights reserved.","generatedAt":"2026-09-21T14:44:19.396Z","games":300000,"seed":20260921,"description":"Distribution of final scores under full-game-optimal Yahtzee play. The policy is exact -- every keep, reroll and category choice is the argmax read off the solved macro-state EV table in yahtzee-fullgame-ev.json, the same decision src/games/dice/lib/fullGamePlay.js makes in the browser -- and only the dice are sampled. Checked against Verhoeff's Optimal Solitaire Yahtzee Player trivia table: the grand total, the standard deviation, the median, and an expected contribution and zero-rate for all thirteen categories plus both bonuses. `histogram` is [score, games] pairs; `cdf` is indexed by score and gives the fraction of games finishing at or below it. Free/laxer joker convention, matching the EV table it plays from.","reference":{"source":"Verhoeff, Optimal Solitaire Yahtzee Player (trivia table)","url":"https://www-set.win.tue.nl/~wstomv/misc/yahtzee/trivia.html","categories":{"ones":{"expected":1.88,"variance":1.48,"zeroPct":10.84},"twos":{"expected":5.28,"variance":3.99,"zeroPct":1.8},"threes":{"expected":8.57,"variance":7.36,"zeroPct":0.95},"fours":{"expected":12.16,"variance":10.8,"zeroPct":0.6},"fives":{"expected":15.69,"variance":14.83,"zeroPct":0.5},"sixes":{"expected":19.19,"variance":21.56,"zeroPct":0.53},"threeKind":{"expected":21.66,"variance":31.56,"zeroPct":3.26},"fourKind":{"expected":13.1,"variance":122.63,"zeroPct":36.34},"fullHouse":{"expected":22.59,"variance":54.41,"zeroPct":9.63},"smallStraight":{"expected":29.46,"variance":15.87,"zeroPct":1.8},"largeStraight":{"expected":32.71,"variance":238.42,"zeroPct":18.22},"yahtzee":{"expected":16.87,"variance":558.88,"zeroPct":66.26},"chance":{"expected":22.01,"variance":6.45,"zeroPct":0}},"upperBonus":{"expected":23.84,"variance":266.04,"zeroPct":31.88},"extraYahtzeeBonus":{"expected":9.58,"variance":1161.19,"zeroPct":91.76},"grandTotal":{"expected":254.5896,"variance":3553.52,"sd":59.6117},"median":248,"yahtzeesRolled":{"expected":0.46,"zeroPct":63.24}}},"summary":{"games":300000,"mean":254.713,"standardError":0.11,"sd":60.241,"median":248,"min":71,"max":749,"upperBonusPct":67.843,"upperBonusExpected":23.745,"yahtzeePct":33.962,"extraBonusPct":8.299,"extraBonusExpected":9.689,"upperBonusVariance":267.25,"extraBonusVariance":1184.4,"variance":3628.97,"yahtzeesRolledMean":0.46,"noYahtzeeRolledPct":63.698,"solvedRootEV":254.6},"percentiles":{"p1":151,"p5":179,"p10":194,"p25":217,"p50":248,"p75":272,"p90":318,"p95":389,"p99":484},"categories":[{"id":"ones","label":"Ones","expected":1.878,"variance":1.46,"zeroPct":10.924,"reference":{"expected":1.88,"variance":1.48,"zeroPct":10.84}},{"id":"twos","label":"Twos","expected":5.267,"variance":3.92,"zeroPct":1.804,"reference":{"expected":5.28,"variance":3.99,"zeroPct":1.8}},{"id":"threes","label":"Threes","expected":8.539,"variance":7.3,"zeroPct":0.978,"reference":{"expected":8.57,"variance":7.36,"zeroPct":0.95}},{"id":"fours","label":"Fours","expected":12.152,"variance":10.76,"zeroPct":0.621,"reference":{"expected":12.16,"variance":10.8,"zeroPct":0.6}},{"id":"fives","label":"Fives","expected":15.682,"variance":14.81,"zeroPct":0.486,"reference":{"expected":15.69,"variance":14.83,"zeroPct":0.5}},{"id":"sixes","label":"Sixes","expected":19.186,"variance":21.53,"zeroPct":0.531,"reference":{"expected":19.19,"variance":21.56,"zeroPct":0.53}},{"id":"threeKind","label":"Three of a Kind","expected":21.692,"variance":31.29,"zeroPct":3.186,"reference":{"expected":21.66,"variance":31.56,"zeroPct":3.26}},{"id":"fourKind","label":"Four of a Kind","expected":13.099,"variance":122.99,"zeroPct":36.465,"reference":{"expected":13.1,"variance":122.63,"zeroPct":36.34}},{"id":"fullHouse","label":"Full House","expected":22.591,"variance":54.42,"zeroPct":9.636,"reference":{"expected":22.59,"variance":54.41,"zeroPct":9.63}},{"id":"smallStraight","label":"Small Straight","expected":29.454,"variance":16.08,"zeroPct":1.82,"reference":{"expected":29.46,"variance":15.87,"zeroPct":1.8}},{"id":"largeStraight","label":"Large 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