{"meta":{"copyright":"© Odds The Law (oddsthelaw.com). Precomputed dataset; all rights reserved.","generatedAt":"2026-09-16T10:26:43.537Z","description":"Per-category detail for Yahtzee, all under the same optimal single-category play as yahtzee-categories.json. `rolls` gives expected score and P(score > 0) with 1, 2 and 3 rolls available, so the value of the two rerolls is explicit. `distribution` is the exact probability of every final score reachable in that category. `bestOpenings`/`worstOpenings` are the initial rolls that leave you best and worst placed, by expected score with two rerolls still to come."},"categories":{"ones":{"rolls":[{"rolls":1,"expectedScore":0.8333,"scorePct":59.8122},{"rolls":2,"expectedScore":1.5278,"scorePct":83.8494},{"rolls":3,"expectedScore":2.1065,"scorePct":93.5095}],"distribution":[{"score":0,"probabilityPct":6.4905},{"score":1,"probabilityPct":23.6256},{"score":2,"probabilityPct":34.3989},{"score":3,"probabilityPct":25.0424},{"score":4,"probabilityPct":9.1154},{"score":5,"probabilityPct":1.3272}],"openingTierCount":6,"topOpeningTiers":[{"ev":5,"probabilityPct":0.0129,"representative":[5,0,0,0,0,0],"immediateScore":5,"rollCount":1},{"ev":4.3056,"probabilityPct":0.3215,"representative":[4,0,0,0,0,1],"immediateScore":4,"rollCount":5},{"ev":3.6111,"probabilityPct":3.215,"representative":[3,0,0,0,1,1],"immediateScore":3,"rollCount":15}],"worstOpeningTier":{"ev":1.5278,"probabilityPct":40.1878,"representative":[0,1,1,1,1,1],"immediateScore":0,"rollCount":126}},"twos":{"rolls":[{"rolls":1,"expectedScore":1.6667,"scorePct":59.8122},{"rolls":2,"expectedScore":3.0556,"scorePct":83.8494},{"rolls":3,"expectedScore":4.213,"scorePct":93.5095}],"distribution":[{"score":0,"probabilityPct":6.4905},{"score":2,"probabilityPct":23.6256},{"score":4,"probabilityPct":34.3989},{"score":6,"probabilityPct":25.0424},{"score":8,"probabilityPct":9.1154},{"score":10,"probabilityPct":1.3272}],"openingTierCount":6,"topOpeningTiers":[{"ev":10,"probabilityPct":0.0129,"representative":[0,5,0,0,0,0],"immediateScore":10,"rollCount":1},{"ev":8.6111,"probabilityPct":0.3215,"representative":[0,4,0,0,0,1],"immediateScore":8,"rollCount":5},{"ev":7.2222,"probabilityPct":3.215,"representative":[0,3,0,0,1,1],"immediateScore":6,"rollCount":15}],"worstOpeningTier":{"ev":3.0556,"probabilityPct":40.1878,"representative":[1,0,1,1,1,1],"immediateScore":0,"rollCount":126}},"threes":{"rolls":[{"rolls":1,"expectedScore":2.5,"scorePct":59.8122},{"rolls":2,"expectedScore":4.5833,"scorePct":83.8494},{"rolls":3,"expectedScore":6.3194,"scorePct":93.5095}],"distribution":[{"score":0,"probabilityPct":6.4905},{"score":3,"probabilityPct":23.6256},{"score":6,"probabilityPct":34.3989},{"score":9,"probabilityPct":25.0424},{"score":12,"probabilityPct":9.1154},{"score":15,"probabilityPct":1.3272}],"openingTierCount":6,"topOpeningTiers":[{"ev":15,"probabilityPct":0.0129,"representative":[0,0,5,0,0,0],"immediateScore":15,"rollCount":1},{"ev":12.9167,"probabilityPct":0.3215,"representative":[0,0,4,0,0,1],"immediateScore":12,"rollCount":5},{"ev":10.8333,"probabilityPct":3.215,"representative":[0,0,3,0,1,1],"immediateScore":9,"rollCount":15}],"worstOpeningTier":{"ev":4.5833,"probabilityPct":40.1878,"representative":[1,1,0,1,1,1],"immediateScore":0,"rollCount":126}},"fours":{"rolls":[{"rolls":1,"expectedScore":3.3333,"scorePct":59.8122},{"rolls":2,"expectedScore":6.1111,"scorePct":83.8494},{"rolls":3,"expectedScore":8.4259,"scorePct":93.5095}],"distribution":[{"score":0,"probabilityPct":6.4905},{"score":4,"probabilityPct":23.6256},{"score":8,"probabilityPct":34.3989},{"score":12,"probabilityPct":25.0424},{"score":16,"probabilityPct":9.1154},{"score":20,"probabilityPct":1.3272}],"openingTierCount":6,"topOpeningTiers":[{"ev":20,"probabilityPct":0.0129,"representative":[0,0,0,5,0,0],"immediateScore":20,"rollCount":1},{"ev":17.2222,"probabilityPct":0.3215,"representative":[0,0,0,4,0,1],"immediateScore":16,"rollCount":5},{"ev":14.4444,"probabilityPct":3.215,"representative":[0,0,0,3,1,1],"immediateScore":12,"rollCount":15}],"worstOpeningTier":{"ev":6.1111,"probabilityPct":40.1878,"representative":[1,1,1,0,1,1],"immediateScore":0,"rollCount":126}},"fives":{"rolls":[{"rolls":1,"expectedScore":4.1667,"scorePct":59.8122},{"rolls":2,"expectedScore":7.6389,"scorePct":83.8494},{"rolls":3,"expectedScore":10.5324,"scorePct":93.5095}],"distribution":[{"score":0,"probabilityPct":6.4905},{"score":5,"probabilityPct":23.6256},{"score":10,"probabilityPct":34.3989},{"score":15,"probabilityPct":25.0424},{"score":20,"probabilityPct":9.1154},{"score":25,"probabilityPct":1.3272}],"openingTierCount":6,"topOpeningTiers":[{"ev":25,"probabilityPct":0.0129,"representative":[0,0,0,0,5,0],"immediateScore":25,"rollCount":1},{"ev":21.5278,"probabilityPct":0.3215,"representative":[0,0,0,0,4,1],"immediateScore":20,"rollCount":5},{"ev":18.0556,"probabilityPct":3.215,"representative":[0,0,0,1,3,1],"immediateScore":15,"rollCount":15}],"worstOpeningTier":{"ev":7.6389,"probabilityPct":40.1878,"representative":[1,1,1,1,0,1],"immediateScore":0,"rollCount":126}},"sixes":{"rolls":[{"rolls":1,"expectedScore":5,"scorePct":59.8122},{"rolls":2,"expectedScore":9.1667,"scorePct":83.8494},{"rolls":3,"expectedScore":12.6389,"scorePct":93.5095}],"distribution":[{"score":0,"probabilityPct":6.4905},{"score":6,"probabilityPct":23.6256},{"score":12,"probabilityPct":34.3989},{"score":18,"probabilityPct":25.0424},{"score":24,"probabilityPct":9.1154},{"score":30,"probabilityPct":1.3272}],"openingTierCount":6,"topOpeningTiers":[{"ev":30,"probabilityPct":0.0129,"representative":[0,0,0,0,0,5],"immediateScore":30,"rollCount":1},{"ev":25.8333,"probabilityPct":0.3215,"representative":[0,0,0,0,1,4],"immediateScore":24,"rollCount":5},{"ev":21.6667,"probabilityPct":3.215,"representative":[0,0,0,1,1,3],"immediateScore":18,"rollCount":15}],"worstOpeningTier":{"ev":9.1667,"probabilityPct":40.1878,"representative":[1,1,1,1,1,0],"immediateScore":0,"rollCount":126}},"threeKind":{"rolls":[{"rolls":1,"expectedScore":3.7269,"scorePct":21.2963},{"rolls":2,"expectedScore":10.2087,"scorePct":53.0607},{"rolls":3,"expectedScore":15.1947,"scorePct":71.2416}],"distribution":[{"score":0,"probabilityPct":28.7584},{"score":5,"probabilityPct":0.0095},{"score":6,"probabilityPct":0.02},{"score":7,"probabilityPct":0.0407},{"score":8,"probabilityPct":0.1171},{"score":9,"probabilityPct":0.2771},{"score":10,"probabilityPct":0.4354},{"score":11,"probabilityPct":0.7579},{"score":12,"probabilityPct":1.1494},{"score":13,"probabilityPct":1.6832},{"score":14,"probabilityPct":2.2175},{"score":15,"probabilityPct":2.405},{"score":16,"probabilityPct":2.9177},{"score":17,"probabilityPct":3.2805},{"score":18,"probabilityPct":3.707},{"score":19,"probabilityPct":4.5137},{"score":20,"probabilityPct":5.0483},{"score":21,"probabilityPct":5.2729},{"score":22,"probabilityPct":5.3972},{"score":23,"probabilityPct":6.0982},{"score":24,"probabilityPct":5.9018},{"score":25,"probabilityPct":5.4144},{"score":26,"probabilityPct":4.5576},{"score":27,"probabilityPct":4.5609},{"score":28,"probabilityPct":3.0718},{"score":29,"probabilityPct":1.7912},{"score":30,"probabilityPct":0.5956}],"openingTierCount":43,"topOpeningTiers":[{"ev":30,"probabilityPct":0.0129,"representative":[0,0,0,0,0,5],"immediateScore":30,"rollCount":1},{"ev":29,"probabilityPct":0.0643,"representative":[0,0,0,0,1,4],"immediateScore":29,"rollCount":1},{"ev":28.25,"probabilityPct":0.2572,"representative":[0,0,0,1,0,4],"immediateScore":28,"rollCount":4}],"worstOpeningTier":{"ev":10.4344,"probabilityPct":0.7716,"representative":[2,1,1,1,0,0],"immediateScore":0,"rollCount":1}},"fourKind":{"rolls":[{"rolls":1,"expectedScore":0.3511,"scorePct":2.0062},{"rolls":2,"expectedScore":2.4215,"scorePct":13.2302},{"rolls":3,"expectedScore":5.6113,"scorePct":27.7443}],"distribution":[{"score":0,"probabilityPct":72.2557},{"score":5,"probabilityPct":0.1596},{"score":6,"probabilityPct":0.2423},{"score":7,"probabilityPct":0.2423},{"score":8,"probabilityPct":0.4344},{"score":9,"probabilityPct":0.892},{"score":10,"probabilityPct":0.7593},{"score":11,"probabilityPct":0.3909},{"score":12,"probabilityPct":0.6633},{"score":13,"probabilityPct":1.3246},{"score":14,"probabilityPct":1.3364},{"score":15,"probabilityPct":0.3586},{"score":16,"probabilityPct":0.958},{"score":17,"probabilityPct":1.7306},{"score":18,"probabilityPct":1.7425},{"score":19,"probabilityPct":0.7058},{"score":20,"probabilityPct":0.6133},{"score":21,"probabilityPct":1.9773},{"score":22,"probabilityPct":1.9892},{"score":23,"probabilityPct":0.8156},{"score":24,"probabilityPct":1.2443},{"score":25,"probabilityPct":1.7524},{"score":26,"probabilityPct":2.3736},{"score":27,"probabilityPct":1.0507},{"score":28,"probabilityPct":1.5424},{"score":29,"probabilityPct":1.6092},{"score":30,"probabilityPct":0.8357}],"openingTierCount":32,"topOpeningTiers":[{"ev":30,"probabilityPct":0.0129,"representative":[0,0,0,0,0,5],"immediateScore":30,"rollCount":1},{"ev":29,"probabilityPct":0.0643,"representative":[0,0,0,0,1,4],"immediateScore":29,"rollCount":1},{"ev":28.25,"probabilityPct":0.2572,"representative":[0,0,0,1,0,4],"immediateScore":28,"rollCount":4}],"worstOpeningTier":{"ev":2.5272,"probabilityPct":0.7716,"representative":[2,1,1,1,0,0],"immediateScore":0,"rollCount":1}},"fullHouse":{"rolls":[{"rolls":1,"expectedScore":0.9645,"scorePct":3.858},{"rolls":2,"expectedScore":4.7806,"scorePct":19.1222},{"rolls":3,"expectedScore":9.0721,"scorePct":36.2883}],"distribution":[{"score":0,"probabilityPct":63.7117},{"score":25,"probabilityPct":36.2883}],"openingTierCount":6,"topOpeningTiers":[{"ev":25,"probabilityPct":3.858,"representative":[0,0,0,0,2,3],"immediateScore":25,"rollCount":30},{"ev":13.8889,"probabilityPct":23.1481,"representative":[0,0,0,1,2,2],"immediateScore":0,"rollCount":60},{"ev":7.6389,"probabilityPct":17.3611,"representative":[0,0,0,1,1,3],"immediateScore":0,"rollCount":90}],"worstOpeningTier":{"ev":4.7806,"probabilityPct":9.2593,"representative":[0,1,1,1,1,1],"immediateScore":0,"rollCount":6}},"smallStraight":{"rolls":[{"rolls":1,"expectedScore":4.6296,"scorePct":15.4321},{"rolls":2,"expectedScore":12.3536,"scorePct":41.1788},{"rolls":3,"expectedScore":18.4633,"scorePct":61.5442}],"distribution":[{"score":0,"probabilityPct":38.4558},{"score":30,"probabilityPct":61.5442}],"openingTierCount":7,"topOpeningTiers":[{"ev":30,"probabilityPct":15.4321,"representative":[0,1,1,1,1,1],"immediateScore":30,"rollCount":16},{"ev":24.0741,"probabilityPct":10.0309,"representative":[0,1,1,1,0,2],"immediateScore":0,"rollCount":20},{"ev":19.7531,"probabilityPct":7.3302,"representative":[1,0,1,1,0,2],"immediateScore":0,"rollCount":20}],"worstOpeningTier":{"ev":12.3536,"probabilityPct":13.1687,"representative":[1,1,0,0,1,2],"immediateScore":0,"rollCount":56}},"largeStraight":{"rolls":[{"rolls":1,"expectedScore":1.2346,"scorePct":3.0864},{"rolls":2,"expectedScore":5.1899,"scorePct":12.9747},{"rolls":3,"expectedScore":10.4438,"scorePct":26.1095}],"distribution":[{"score":0,"probabilityPct":73.8905},{"score":40,"probabilityPct":26.1095}],"openingTierCount":7,"topOpeningTiers":[{"ev":40,"probabilityPct":3.0864,"representative":[0,1,1,1,1,1],"immediateScore":40,"rollCount":2},{"ev":22.2222,"probabilityPct":3.0864,"representative":[0,1,1,1,2,0],"immediateScore":0,"rollCount":4},{"ev":12.2222,"probabilityPct":30.8642,"representative":[1,0,1,1,1,1],"immediateScore":0,"rollCount":36}],"worstOpeningTier":{"ev":5.1899,"probabilityPct":0.4115,"representative":[2,0,0,0,0,3],"immediateScore":0,"rollCount":6}},"yahtzee":{"rolls":[{"rolls":1,"expectedScore":0.0386,"scorePct":0.0772},{"rolls":2,"expectedScore":0.6316,"scorePct":1.2631},{"rolls":3,"expectedScore":2.3014,"scorePct":4.6029}],"distribution":[{"score":0,"probabilityPct":95.3971},{"score":50,"probabilityPct":4.6029}],"openingTierCount":5,"topOpeningTiers":[{"ev":50,"probabilityPct":0.0772,"representative":[0,0,0,0,0,5],"immediateScore":50,"rollCount":6},{"ev":15.2778,"probabilityPct":1.929,"representative":[0,0,0,0,1,4],"immediateScore":0,"rollCount":30},{"ev":4.6682,"probabilityPct":19.2901,"representative":[0,0,0,1,1,3],"immediateScore":0,"rollCount":90}],"worstOpeningTier":{"ev":0.6316,"probabilityPct":9.2593,"representative":[0,1,1,1,1,1],"immediateScore":0,"rollCount":6}},"chance":{"rolls":[{"rolls":1,"expectedScore":17.5,"scorePct":100},{"rolls":2,"expectedScore":21.25,"scorePct":100},{"rolls":3,"expectedScore":23.3333,"scorePct":100}],"distribution":[{"score":5,"probabilityPct":0.0001},{"score":6,"probabilityPct":0.0003},{"score":7,"probabilityPct":0.0008},{"score":8,"probabilityPct":0.0024},{"score":9,"probabilityPct":0.0071},{"score":10,"probabilityPct":0.0186},{"score":11,"probabilityPct":0.0421},{"score":12,"probabilityPct":0.0913},{"score":13,"probabilityPct":0.1929},{"score":14,"probabilityPct":0.3781},{"score":15,"probabilityPct":0.6827},{"score":16,"probabilityPct":1.1757},{"score":17,"probabilityPct":1.9584},{"score":18,"probabilityPct":3.0761},{"score":19,"probabilityPct":4.4891},{"score":20,"probabilityPct":6.1952},{"score":21,"probabilityPct":8.2066},{"score":22,"probabilityPct":10.2452},{"score":23,"probabilityPct":11.717},{"score":24,"probabilityPct":12.2885},{"score":25,"probabilityPct":12.0713},{"score":26,"probabilityPct":10.9739},{"score":27,"probabilityPct":8.5734},{"score":28,"probabilityPct":5.144},{"score":29,"probabilityPct":2.0576},{"score":30,"probabilityPct":0.4115}],"openingTierCount":21,"topOpeningTiers":[{"ev":30,"probabilityPct":0.0129,"representative":[0,0,0,0,0,5],"immediateScore":30,"rollCount":1},{"ev":29,"probabilityPct":0.0643,"representative":[0,0,0,0,1,4],"immediateScore":29,"rollCount":1},{"ev":28.25,"probabilityPct":0.2572,"representative":[0,0,0,1,0,4],"immediateScore":28,"rollCount":4}],"worstOpeningTier":{"ev":21.25,"probabilityPct":13.1687,"representative":[1,1,1,2,0,0],"immediateScore":14,"rollCount":56}}}}