{ "license": "CC BY 4.0 — https://creativecommons.org/licenses/by/4.0/", "attribution": "Wheel Cue (wheelcue.com)", "source": "https://wheelcue.com/research/are-random-teams-balanced/", "slug": "are-random-teams-balanced", "generatedAt": "2026-10-04T14:19:57.962Z", "node": "v24.21.0", "randomJsSha256": "946e82a3c426d698779d3246c37c732f771db3873c93239ef3952aea39be1e2d", "totalSpins": 62607658, "rawDraws": 62607658, "totalSplits": 3974246, "checksTotal": 99, "checksPassed": 99, "checksSimulatedVsExact": 97, "checksSanity": 2, "runtimeSeconds": 111.6, "params": { "classSize": 24, "teamCounts": [ 2, 3, 4, 6 ], "splitsPerTeamCount": { "2": 400000, "3": 200000, "4": 800000, "6": 400000 }, "skill": { "ranks": "Every student has a distinct rank 1..24 (24 = strongest). Team strength = sum of ranks.", "ratingScale": "1 (weakest) to 5 (strongest)", "ratingCounts": { "1": 3, "2": 5, "3": 8, "4": 5, "5": 3 }, "ratingByRank": "ranks 1-3 -> 1, 4-8 -> 2, 9-16 -> 3, 17-21 -> 4, 22-24 -> 5 (the rating is a coarser view of the same order)", "closeGapPerPlayer": 0.5, "lopsidedGapPerPlayer": 1 }, "topPlayers": { "tops": [ 2, 3, 4, 5, 6, 8 ] }, "composition": { "groupA": 12, "groupB": 12, "note": "two groups of 12, e.g. 12 girls and 12 boys" }, "friends": { "pairs": 5, "pairsTable": [ 1, 2, 3, 4, 5, 6, 8, 10, 12 ], "note": "disjoint pairs of friends; 'together' = both on the same team" }, "tiers": { "tierSize": 8, "tiers": [ "top 8 (ranks 17-24)", "middle 8 (ranks 9-16)", "bottom 8 (ranks 1-8)" ], "teamCounts": [ 2, 4 ], "splits": 300000 }, "swaps": { "teamCounts": [ 2, 4 ], "maxSwaps": 2 }, "sizeMode": { "peoplePerTeam": 6, "splits": 100000 }, "regenerate": { "teams": 4, "maxGapTotal": 3, "experiments": 40000 }, "widgetDefaults": { "classSize": 24, "teams": 4, "topPlayers": 4, "groupA": 12, "splits": 5000 } }, "notes": { "model": "The tool (app.js initTeams) shuffles with CueRandom.shuffle (Fisher–Yates, 23 draws for 24 names) and deals round-robin: Team j gets shuffled positions j, j+k, j+2k, ... Every assignment of the 24 students to labeled teams of 24/k is equally likely, so any fixed labelling of who is 'top', 'group A' or a 'friend pair' gives the same probabilities. 'People per team' mode with 6 per team gives the same distribution (checked).", "totalSpins": "totalSpins = CueRandom draws that place a name (L - 1 per shuffle of L names); rawDraws also counts draws rejected for modulo bias. totalSplits = presses of Generate simulated (a tiered split is three presses of 8 names). In the article, quote totalSplits as the number of simulated presses of Generate (team splits) and call totalSpins 'random draws', not spins.", "simulatedVsExact": "results holds simulated splits (never site usage), including the simulated columns of every chart (results.*.distribution, results.composition.byTeams[k].mostLopsidedTeam), matched to theory rows by key; theory holds exact values only and is the same on every run. Statements in the article use theory; simulated values are called 'simulated'. Quantities marked simulationOnly (rank gaps with 4 teams, rank hand swaps with 4 teams) have no exact counterpart and are not checked.", "rounding": "Probabilities are rounded to 6 decimals. Use the display strings for percentages; never print a share as 100% or 0% unless it is exactly that.", "checks": "Every check of kind 'simulated vs exact' allows 4 standard errors of the simulated share or average (se from the exact value, never below one split's worth for a share); shares within 0.0001 of 0 or 1 are not checked. Checks of kind 'sanity' (se 0) confirm the simulation's own bookkeeping and must hold exactly. Describe them as checksSimulatedVsExact simulated-vs-exact checks plus checksSanity sanity checks." }, "display": { "twoBestTogether4Teams": "5/23 (21.7%)", "twoBestTogether2Teams": "11/23 (47.8%)", "ratingsClose4Teams": "14.6%", "ratingsLopsided4Teams": "58.2%", "ratingsClose2Teams": "75.6%", "ratingsLopsided2Teams": "6.3%", "topFourSpread4Teams": "12.2%", "topFourSomeTeam3Plus4Teams": "14.1%", "compositionAllEven4Teams": "5.9%", "compositionLopsided4Teams": "42.9%", "friendsAtLeastOne4Teams": "70.3%", "tieredClose4Teams": "93.9%", "oneSwapClose4Teams": "77.1%", "twoSwapsClose4Teams": "99.33%", "regenerateMeanPresses": "6.8", "regenerateOneIn": "about 1 in 7" }, "theory": { "ratings": { "note": "Gap = strongest team's total rating minus weakest team's. gapPerPlayer = gap / team size = difference between the two teams' AVERAGE ratings. 'close' = gapPerPlayer <= 0.5; 'lopsided' = gapPerPlayer >= 1. Exact: DP over teams on how many students of each rating each team gets.", "byTeams": { "2": { "teams": 2, "teamSize": 12, "closeMaxGap": 6, "lopsidedMinGap": 12, "pClose": 0.755877, "pLopsided": 0.062514, "pPerfectlyEven": 0.130726, "meanGap": 4.7473, "sdGap": 3.5974, "meanGapPerPlayer": 0.3956, "medianGap": 4, "p90Gap": 10, "expectedTriesForClose": 1.323, "distribution": [ { "gap": 0, "gapPerPlayer": 0, "exact": 0.130726 }, { "gap": 2, "gapPerPlayer": 0.1667, "exact": 0.248511 }, { "gap": 4, "gapPerPlayer": 0.3333, "exact": 0.21273 }, { "gap": 6, "gapPerPlayer": 0.5, "exact": 0.163909 }, { "gap": 8, "gapPerPlayer": 0.6667, "exact": 0.112735 }, { "gap": 10, "gapPerPlayer": 0.8333, "exact": 0.068875 }, { "gap": 12, "gapPerPlayer": 1, "exact": 0.036758 }, { "gap": 14, "gapPerPlayer": 1.1667, "exact": 0.016898 }, { "gap": 16, "gapPerPlayer": 1.3333, "exact": 0.006435 }, { "gap": 18, "gapPerPlayer": 1.5, "exact": 0.001957 }, { "gap": 20, "gapPerPlayer": 1.6667, "exact": 0.000414 }, { "gap": 22, "gapPerPlayer": 1.8333, "exact": 0.000052 } ], "display": { "pClose": "75.6%", "pLopsided": "6.3%", "pPerfectlyEven": "13.1%" } }, "3": { "teams": 3, "teamSize": 8, "closeMaxGap": 4, "lopsidedMinGap": 8, "pClose": 0.34395, "pLopsided": 0.285661, "pPerfectlyEven": 0.02169, "meanGap": 5.8523, "sdGap": 2.9731, "meanGapPerPlayer": 0.7315, "medianGap": 6, "p90Gap": 10, "expectedTriesForClose": 2.907, "distribution": [ { "gap": 0, "gapPerPlayer": 0, "exact": 0.02169 }, { "gap": 2, "gapPerPlayer": 0.25, "exact": 0.121128 }, { "gap": 3, "gapPerPlayer": 0.375, "exact": 0.104418 }, { "gap": 4, "gapPerPlayer": 0.5, "exact": 0.096715 }, { "gap": 5, "gapPerPlayer": 0.625, "exact": 0.154538 }, { "gap": 6, "gapPerPlayer": 0.75, "exact": 0.118735 }, { "gap": 7, "gapPerPlayer": 0.875, "exact": 0.097117 }, { "gap": 8, "gapPerPlayer": 1, "exact": 0.098302 }, { "gap": 9, "gapPerPlayer": 1.125, "exact": 0.066347 }, { "gap": 10, "gapPerPlayer": 1.25, "exact": 0.046394 }, { "gap": 11, "gapPerPlayer": 1.375, "exact": 0.034301 }, { "gap": 12, "gapPerPlayer": 1.5, "exact": 0.019514 }, { "gap": 13, "gapPerPlayer": 1.625, "exact": 0.011013 }, { "gap": 14, "gapPerPlayer": 1.75, "exact": 0.005839 }, { "gap": 15, "gapPerPlayer": 1.875, "exact": 0.002549 }, { "gap": 16, "gapPerPlayer": 2, "exact": 0.000995 }, { "gap": 17, "gapPerPlayer": 2.125, "exact": 0.000315 }, { "gap": 18, "gapPerPlayer": 2.25, "exact": 0.000077 }, { "gap": 19, "gapPerPlayer": 2.375, "exact": 0.000013 } ], "display": { "pClose": "34.4%", "pLopsided": "28.6%", "pPerfectlyEven": "2.2%" } }, "4": { "teams": 4, "teamSize": 6, "closeMaxGap": 3, "lopsidedMinGap": 6, "pClose": 0.14605, "pLopsided": 0.582007, "pPerfectlyEven": 0.004193, "meanGap": 6.1592, "sdGap": 2.4969, "meanGapPerPlayer": 1.0265, "medianGap": 6, "p90Gap": 9, "expectedTriesForClose": 6.847, "distribution": [ { "gap": 0, "gapPerPlayer": 0, "exact": 0.004193 }, { "gap": 2, "gapPerPlayer": 0.3333, "exact": 0.066322 }, { "gap": 3, "gapPerPlayer": 0.5, "exact": 0.075536 }, { "gap": 4, "gapPerPlayer": 0.6667, "exact": 0.127221 }, { "gap": 5, "gapPerPlayer": 0.8333, "exact": 0.144722 }, { "gap": 6, "gapPerPlayer": 1, "exact": 0.15693 }, { "gap": 7, "gapPerPlayer": 1.1667, "exact": 0.135257 }, { "gap": 8, "gapPerPlayer": 1.3333, "exact": 0.11051 }, { "gap": 9, "gapPerPlayer": 1.5, "exact": 0.079336 }, { "gap": 10, "gapPerPlayer": 1.6667, "exact": 0.0506 }, { "gap": 11, "gapPerPlayer": 1.8333, "exact": 0.028102 }, { "gap": 12, "gapPerPlayer": 2, "exact": 0.013553 }, { "gap": 13, "gapPerPlayer": 2.1667, "exact": 0.005443 }, { "gap": 14, "gapPerPlayer": 2.3333, "exact": 0.001753 }, { "gap": 15, "gapPerPlayer": 2.5, "exact": 0.000435 }, { "gap": 16, "gapPerPlayer": 2.6667, "exact": 0.000078 } ], "display": { "pClose": "14.6%", "pLopsided": "58.2%", "pPerfectlyEven": "0.42%" } }, "6": { "teams": 6, "teamSize": 4, "closeMaxGap": 2, "lopsidedMinGap": 4, "pClose": 0.022671, "pLopsided": 0.928567, "pPerfectlyEven": 0.0002, "meanGap": 6.1668, "sdGap": 1.8825, "meanGapPerPlayer": 1.5417, "medianGap": 6, "p90Gap": 9, "expectedTriesForClose": 44.11, "distribution": [ { "gap": 0, "gapPerPlayer": 0, "exact": 0.0002 }, { "gap": 2, "gapPerPlayer": 0.5, "exact": 0.022471 }, { "gap": 3, "gapPerPlayer": 0.75, "exact": 0.048762 }, { "gap": 4, "gapPerPlayer": 1, "exact": 0.122287 }, { "gap": 5, "gapPerPlayer": 1.25, "exact": 0.178845 }, { "gap": 6, "gapPerPlayer": 1.5, "exact": 0.203853 }, { "gap": 7, "gapPerPlayer": 1.75, "exact": 0.18293 }, { "gap": 8, "gapPerPlayer": 2, "exact": 0.129764 }, { "gap": 9, "gapPerPlayer": 2.25, "exact": 0.071176 }, { "gap": 10, "gapPerPlayer": 2.5, "exact": 0.029144 }, { "gap": 11, "gapPerPlayer": 2.75, "exact": 0.008605 }, { "gap": 12, "gapPerPlayer": 3, "exact": 0.001728 }, { "gap": 13, "gapPerPlayer": 3.25, "exact": 0.000221 }, { "gap": 14, "gapPerPlayer": 3.5, "exact": 0.000015 } ], "display": { "pClose": "2.3%", "pLopsided": "92.9%", "pPerfectlyEven": "0.02%" } } } }, "ranks": { "note": "Ranks 1..24, all different (24 = strongest). Gap in rank points; gapPerPlayer = gap / team size = difference between team AVERAGE ranks, in places. pGapAtMostTeamSize = P(gap <= team size) = team average ranks within 1 place. 2 teams: exact (subset-sum DP over the C(24,12) = 2,704,156 possible Team 1s). 4 teams: the gap is simulation only (no tractable exact method used; see results.ranks.fourTeams); Team 1's total is exact (6-subset DP) and checked instead.", "twoTeams": { "meanGap": 27.8115, "sdGap": 20.6524, "meanGapPerPlayer": 2.3176, "medianGap": 24, "p90Gap": 58, "teamSize": 12, "pGapAtMostTeamSize": 0.287465, "pGapAtLeast36": 0.318585, "maxPossibleGap": 144, "distributionBy12": [ { "from": 0, "to": 11, "exact": 0.244715 }, { "from": 12, "to": 23, "exact": 0.241562 }, { "from": 24, "to": 35, "exact": 0.195137 }, { "from": 36, "to": 47, "exact": 0.140403 }, { "from": 48, "to": 59, "exact": 0.089448 }, { "from": 60, "to": 71, "exact": 0.050013 }, { "from": 72, "to": 83, "exact": 0.024229 }, { "from": 84, "to": 95, "exact": 0.009979 }, { "from": 96, "to": 107, "exact": 0.003398 }, { "from": 108, "to": 119, "exact": 0.000913 }, { "from": 120, "to": 131, "exact": 0.000179 }, { "from": 132, "to": 143, "exact": 0.000021 }, { "from": 144, "to": 155, "exact": 0.000001 } ] }, "fourTeams": { "team1Total": { "mean": 75, "sd": 15, "pAtLeast90": 0.171617, "pAtMost60": 0.171617, "note": "Team 1 = a uniformly random 6 of the 24 ranks (round-robin positions 1, 5, 9, ...). Mean 75, variance 6 * (24^2 - 1)/12 * 18/23 = 225." }, "gapSimulationOnly": true } }, "tiered": { "note": "Split the class into tiers by skill (top 8 = ranks 17-24, middle 8 = ranks 9-16, bottom 8 = ranks 1-8; with the rating mix the top 8 are the three 5s and five 4s, the middle 8 are all 3s, the bottom 8 are five 2s and three 1s). Run the generator on each tier with the same number of teams and combine Team 1s, Team 2s, ... Each tier of 8 deals evenly (8/k per team) only because 8 is a multiple of k; otherwise Team 1 would get the extra person from every tier. Exact by enumerating every deal of each tier.", "byTeams": { "2": { "teams": 2, "teamSize": 12, "ratings": { "pClose": 1, "pLopsided": 0, "pPerfectlyEven": 0.377551, "meanGap": 1.5306, "sdGap": 1.3939, "maxGap": 6, "distribution": [ { "gap": 0, "exact": 0.377551 }, { "gap": 2, "exact": 0.489796 }, { "gap": 4, "exact": 0.122449 }, { "gap": 6, "exact": 0.010204 } ] }, "ranks": { "meanGap": 9.6164, "sdGap": 7.1781, "meanGapPerPlayer": 0.8014, "medianGap": 8, "p90Gap": 20, "pGapAtMostTeamSize": 0.715942, "maxGap": 48 } }, "4": { "teams": 4, "teamSize": 6, "ratings": { "pClose": 0.938776, "pLopsided": 0, "pPerfectlyEven": 0.096939, "meanGap": 2.1122, "sdGap": 0.8965, "maxGap": 4, "distribution": [ { "gap": 0, "exact": 0.096939 }, { "gap": 2, "exact": 0.658163 }, { "gap": 3, "exact": 0.183673 }, { "gap": 4, "exact": 0.061224 } ] }, "ranks": { "meanGap": 12.4359, "sdGap": 5.0261, "meanGapPerPlayer": 2.0726, "medianGap": 12, "p90Gap": 19, "pGapAtMostTeamSize": 0.120132, "maxGap": 36 } } } }, "swaps": { "note": "After a random split, trade one player between two teams: a stronger player from the stronger team for a weaker one from the weaker team. Of all such trades (any two teams, any two players) the one leaving the smallest gap between the strongest and weakest team is made; if several do, the one that brings the team totals closest together (smallest sum of squared team totals); then the smaller skill moved. A trade is made only if it improves on doing nothing by that score, so when two teams tie for strongest or weakest the rule evens the totals out instead of stopping. Row 'swaps: 2' applies the rule a second time. Ratings: exact, by applying the rule to every possible labeled team composition (weights = number of splits). Ranks: exact for 2 teams (the rule applied to all C(24,12) = 2,704,156 possible Team 1s); simulation only for 4 teams (see results.swaps.byTeams[4].ranks). With 2 teams the sum of the two totals is fixed, so the tie-break never matters there.", "byTeams": { "2": { "teams": 2, "teamSize": 12, "ratings": [ { "swaps": 0, "pClose": 0.755877, "pLopsided": 0.062514, "pPerfectlyEven": 0.130726, "meanGap": 4.7473, "sdGap": 3.5974, "maxGap": 22 }, { "swaps": 1, "pClose": 0.991143, "pLopsided": 0.000466, "pPerfectlyEven": 0.868397, "meanGap": 0.4634, "sdGap": 1.3912, "maxGap": 14 }, { "swaps": 2, "pClose": 1, "pLopsided": 0, "pPerfectlyEven": 0.997577, "meanGap": 0.0059, "sdGap": 0.1276, "maxGap": 6 } ], "ranks": [ { "swaps": 0, "meanGap": 27.8115, "sdGap": 20.6524, "meanGapPerPlayer": 2.3176, "pGapAtMostTeamSize": 0.287465, "pPerfectlyEven": 0.022598, "maxGap": 144 }, { "swaps": 1, "meanGap": 3.1048, "sdGap": 8.4988, "meanGapPerPlayer": 0.2587, "pGapAtMostTeamSize": 0.905463, "pPerfectlyEven": 0.805442, "maxGap": 98 }, { "swaps": 2, "meanGap": 0.0936, "sdGap": 1.2219, "meanGapPerPlayer": 0.0078, "pGapAtMostTeamSize": 0.997446, "pPerfectlyEven": 0.990459, "maxGap": 56 } ], "ranksSimulationOnly": false }, "4": { "teams": 4, "teamSize": 6, "ratings": [ { "swaps": 0, "pClose": 0.14605, "pLopsided": 0.582007, "pPerfectlyEven": 0.004193, "meanGap": 6.1592, "sdGap": 2.4969, "maxGap": 18 }, { "swaps": 1, "pClose": 0.771035, "pLopsided": 0.04278, "pPerfectlyEven": 0.114825, "meanGap": 2.5711, "sdGap": 1.4883, "maxGap": 10 }, { "swaps": 2, "pClose": 0.993261, "pLopsided": 0.0001, "pPerfectlyEven": 0.596685, "meanGap": 0.8356, "sdGap": 1.0373, "maxGap": 6 } ], "ranks": null, "ranksSimulationOnly": true } } }, "regenerate": { "note": "Press Generate until the 4-team gap is at most 3 rating points (team averages within 0.5). Tries are geometric with success chance p per press: mean 1/p, P(done within m presses) = 1 - (1-p)^m. 'withTopFourSpread' asks for the close gap AND the four best students (ranks 21-24) on four different teams, exact by enumerating compositions with rank 21 tracked separately. meanGapWhenDone = the expected gap of the split you finally keep, E[gap | gap <= 3].", "pPerTry": 0.14605, "meanTries": 6.847, "sdTries": 6.327, "medianTries": 5, "p90Tries": 15, "pWithin3": 0.377273, "pWithin5": 0.545888, "pWithin10": 0.793782, "pMoreThan10": 0.206218, "meanGapWhenDone": 2.4598, "sdGapWhenDone": 0.6486, "gapWhenDone": [ { "gap": 0, "p": 0.028706 }, { "gap": 2, "p": 0.454103 }, { "gap": 3, "p": 0.517191 } ], "withTopFourSpread": { "pPerTry": 0.034601, "pPerTryFraction": "19812750/572605033", "meanTries": 28.901, "pWithin10": 0.296817, "pTopFourSpreadAlone": 0.121965 }, "topFourSpreadOnly": { "pPerTry": 0.121965, "meanTries": 8.199 }, "display": { "pPerTry": "14.6%", "pWithin3": "37.7%", "pMoreThan10": "20.6%" } }, "topPlayers": { "note": "How the t strongest students fall across the teams: multivariate hypergeometric, P(c_1..c_k) = prod C(s_j, c_j) / C(24, t). 'asEvenAsPossible' = every team has floor(t/k) or ceil(t/k) of them. P(the two best together) = (s-1)/(n-1).", "byTeams": { "2": { "teams": 2, "teamSize": 12, "pTwoBestTogether": 0.478261, "pTwoBestTogetherFraction": "11/23", "byTop": [ { "top": 2, "pSomeTeamNone": 0.478261, "pAsEvenAsPossible": 0.521739, "asEvenFraction": "12/23", "pSomeTeam3Plus": 0, "expectedMostOnOneTeam": 1.4783, "mostOnOneTeam": [ { "most": 1, "p": 0.521739 }, { "most": 2, "p": 0.478261 } ] }, { "top": 3, "pSomeTeamNone": 0.217391, "pAsEvenAsPossible": 0.782609, "asEvenFraction": "18/23", "pSomeTeam3Plus": 0.217391, "expectedMostOnOneTeam": 2.2174, "mostOnOneTeam": [ { "most": 2, "p": 0.782609 }, { "most": 3, "p": 0.217391 } ] }, { "top": 4, "pSomeTeamNone": 0.093168, "pAsEvenAsPossible": 0.409938, "asEvenFraction": "66/161", "pSomeTeam3Plus": 0.590062, "expectedMostOnOneTeam": 2.6832, "mostOnOneTeam": [ { "most": 2, "p": 0.409938 }, { "most": 3, "p": 0.496894 }, { "most": 4, "p": 0.093168 } ] }, { "top": 5, "pSomeTeamNone": 0.037267, "pAsEvenAsPossible": 0.68323, "asEvenFraction": "110/161", "pSomeTeam3Plus": 1, "expectedMostOnOneTeam": 3.354, "mostOnOneTeam": [ { "most": 3, "p": 0.68323 }, { "most": 4, "p": 0.279503 }, { "most": 5, "p": 0.037267 } ] }, { "top": 6, "pSomeTeamNone": 0.01373, "pAsEvenAsPossible": 0.359595, "asEvenFraction": "1100/3059", "pSomeTeam3Plus": 1, "expectedMostOnOneTeam": 3.8091, "mostOnOneTeam": [ { "most": 3, "p": 0.359595 }, { "most": 4, "p": 0.485453 }, { "most": 5, "p": 0.141223 }, { "most": 6, "p": 0.01373 } ] }, { "top": 8, "pSomeTeamNone": 0.001346, "pAsEvenAsPossible": 0.333154, "asEvenFraction": "2475/7429", "pSomeTeam3Plus": 1, "expectedMostOnOneTeam": 4.8884, "mostOnOneTeam": [ { "most": 4, "p": 0.333154 }, { "most": 5, "p": 0.473819 }, { "most": 6, "p": 0.165837 }, { "most": 7, "p": 0.025845 }, { "most": 8, "p": 0.001346 } ] } ] }, "3": { "teams": 3, "teamSize": 8, "pTwoBestTogether": 0.304348, "pTwoBestTogetherFraction": "7/23", "byTop": [ { "top": 2, "pSomeTeamNone": 1, "pAsEvenAsPossible": 0.695652, "asEvenFraction": "16/23", "pSomeTeam3Plus": 0, "expectedMostOnOneTeam": 1.3043, "mostOnOneTeam": [ { "most": 1, "p": 0.695652 }, { "most": 2, "p": 0.304348 } ] }, { "top": 3, "pSomeTeamNone": 0.747036, "pAsEvenAsPossible": 0.252964, "asEvenFraction": "64/253", "pSomeTeam3Plus": 0.083004, "expectedMostOnOneTeam": 1.83, "mostOnOneTeam": [ { "most": 1, "p": 0.252964 }, { "most": 2, "p": 0.664032 }, { "most": 3, "p": 0.083004 } ] }, { "top": 4, "pSomeTeamNone": 0.494071, "pAsEvenAsPossible": 0.505929, "asEvenFraction": "128/253", "pSomeTeam3Plus": 0.272727, "expectedMostOnOneTeam": 2.2925, "mostOnOneTeam": [ { "most": 2, "p": 0.727273 }, { "most": 3, "p": 0.252964 }, { "most": 4, "p": 0.019763 } ] }, { "top": 5, "pSomeTeamNone": 0.304348, "pAsEvenAsPossible": 0.442688, "asEvenFraction": "112/253", "pSomeTeam3Plus": 0.557312, "expectedMostOnOneTeam": 2.6443, "mostOnOneTeam": [ { "most": 2, "p": 0.442688 }, { "most": 3, "p": 0.474308 }, { "most": 4, "p": 0.079051 }, { "most": 5, "p": 0.003953 } ] }, { "top": 6, "pSomeTeamNone": 0.177866, "pAsEvenAsPossible": 0.163095, "asEvenFraction": "784/4807", "pSomeTeam3Plus": 0.836905, "expectedMostOnOneTeam": 3.0659, "mostOnOneTeam": [ { "most": 2, "p": 0.163095 }, { "most": 3, "p": 0.629083 }, { "most": 4, "p": 0.187227 }, { "most": 5, "p": 0.019971 }, { "most": 6, "p": 0.000624 } ] }, { "top": 8, "pSomeTeamNone": 0.052493, "pAsEvenAsPossible": 0.35817, "asEvenFraction": "87808/245157", "pSomeTeam3Plus": 1, "expectedMostOnOneTeam": 3.7987, "mostOnOneTeam": [ { "most": 3, "p": 0.35817 }, { "most": 4, "p": 0.49968 }, { "most": 5, "p": 0.127918 }, { "most": 6, "p": 0.013706 }, { "most": 7, "p": 0.000522 }, { "most": 8, "p": 0.000004 } ] } ] }, "4": { "teams": 4, "teamSize": 6, "pTwoBestTogether": 0.217391, "pTwoBestTogetherFraction": "5/23", "byTop": [ { "top": 2, "pSomeTeamNone": 1, "pAsEvenAsPossible": 0.782609, "asEvenFraction": "18/23", "pSomeTeam3Plus": 0, "expectedMostOnOneTeam": 1.2174, "mostOnOneTeam": [ { "most": 1, "p": 0.782609 }, { "most": 2, "p": 0.217391 } ] }, { "top": 3, "pSomeTeamNone": 1, "pAsEvenAsPossible": 0.426877, "asEvenFraction": "108/253", "pSomeTeam3Plus": 0.039526, "expectedMostOnOneTeam": 1.6126, "mostOnOneTeam": [ { "most": 1, "p": 0.426877 }, { "most": 2, "p": 0.533597 }, { "most": 3, "p": 0.039526 } ] }, { "top": 4, "pSomeTeamNone": 0.878035, "pAsEvenAsPossible": 0.121965, "asEvenFraction": "216/1771", "pSomeTeam3Plus": 0.141163, "expectedMostOnOneTeam": 2.0248, "mostOnOneTeam": [ { "most": 1, "p": 0.121965 }, { "most": 2, "p": 0.736872 }, { "most": 3, "p": 0.135517 }, { "most": 4, "p": 0.005647 } ] }, { "top": 5, "pSomeTeamNone": 0.695088, "pAsEvenAsPossible": 0.304912, "asEvenFraction": "540/1771", "pSomeTeam3Plus": 0.313947, "expectedMostOnOneTeam": 2.3405, "mostOnOneTeam": [ { "most": 2, "p": 0.686053 }, { "most": 3, "p": 0.287973 }, { "most": 4, "p": 0.025409 }, { "most": 5, "p": 0.000565 } ] }, { "top": 6, "pSomeTeamNone": 0.510535, "pAsEvenAsPossible": 0.361081, "asEvenFraction": "12150/33649", "pSomeTeam3Plus": 0.538619, "expectedMostOnOneTeam": 2.6133, "mostOnOneTeam": [ { "most": 2, "p": 0.461381 }, { "most": 3, "p": 0.467176 }, { "most": 4, "p": 0.068204 }, { "most": 5, "p": 0.00321 }, { "most": 6, "p": 0.00003 } ] }, { "top": 8, "pSomeTeamNone": 0.233948, "pAsEvenAsPossible": 0.068833, "asEvenFraction": "5625/81719", "pSomeTeam3Plus": 0.931167, "expectedMostOnOneTeam": 3.2347, "mostOnOneTeam": [ { "most": 2, "p": 0.068833 }, { "most": 3, "p": 0.655906 }, { "most": 4, "p": 0.2478 }, { "most": 5, "p": 0.026628 }, { "most": 6, "p": 0.000832 } ] } ] }, "6": { "teams": 6, "teamSize": 4, "pTwoBestTogether": 0.130435, "pTwoBestTogetherFraction": "3/23", "byTop": [ { "top": 2, "pSomeTeamNone": 1, "pAsEvenAsPossible": 0.869565, "asEvenFraction": "20/23", "pSomeTeam3Plus": 0, "expectedMostOnOneTeam": 1.1304, "mostOnOneTeam": [ { "most": 1, "p": 0.869565 }, { "most": 2, "p": 0.130435 } ] }, { "top": 3, "pSomeTeamNone": 1, "pAsEvenAsPossible": 0.632411, "asEvenFraction": "160/253", "pSomeTeam3Plus": 0.011858, "expectedMostOnOneTeam": 1.3794, "mostOnOneTeam": [ { "most": 1, "p": 0.632411 }, { "most": 2, "p": 0.355731 }, { "most": 3, "p": 0.011858 } ] }, { "top": 4, "pSomeTeamNone": 1, "pAsEvenAsPossible": 0.361378, "asEvenFraction": "640/1771", "pSomeTeam3Plus": 0.045737, "expectedMostOnOneTeam": 1.6849, "mostOnOneTeam": [ { "most": 1, "p": 0.361378 }, { "most": 2, "p": 0.592885 }, { "most": 3, "p": 0.045172 }, { "most": 4, "p": 0.000565 } ] }, { "top": 5, "pSomeTeamNone": 1, "pAsEvenAsPossible": 0.144551, "asEvenFraction": "256/1771", "pSomeTeam3Plus": 0.110107, "expectedMostOnOneTeam": 1.9684, "mostOnOneTeam": [ { "most": 1, "p": 0.144551 }, { "most": 2, "p": 0.745342 }, { "most": 3, "p": 0.107284 }, { "most": 4, "p": 0.002823 } ] }, { "top": 6, "pSomeTeamNone": 0.969568, "pAsEvenAsPossible": 0.030432, "asEvenFraction": "1024/33649", "pSomeTeam3Plus": 0.209962, "expectedMostOnOneTeam": 2.188, "mostOnOneTeam": [ { "most": 1, "p": 0.030432 }, { "most": 2, "p": 0.759607 }, { "most": 3, "p": 0.201492 }, { "most": 4, "p": 0.00847 } ] }, { "top": 8, "pSomeTeamNone": 0.778623, "pAsEvenAsPossible": 0.187961, "asEvenFraction": "15360/81719", "pSomeTeam3Plus": 0.503665, "expectedMostOnOneTeam": 2.5432, "mostOnOneTeam": [ { "most": 2, "p": 0.496335 }, { "most": 3, "p": 0.46416 }, { "most": 4, "p": 0.039505 } ] } ] } } }, "composition": { "note": "Two groups of 12 (e.g. girls and boys). mostLopsidedTeam = the largest single-group count on any team (e.g. 5 means some team is 5-1 or 6-0 when teams are 6). Exact: multivariate hypergeometric over how group A's 12 fall across the teams. pSomeTeamLopsided uses a cut that is set per team size (lopsidedFrom: 8-4 for teams of 12, 6-2 for 8, 5-1 for 6, 4-0 for 4), so it must always be quoted with its lopsidedLabel and NEVER compared across team counts. For a comparison across team counts use pSomeTeamThreeQuarters: some team has at least three quarters of its members from one group (9-3, 6-2, 5-1, 3-1).", "byTeams": { "2": { "teams": 2, "teamSize": 12, "pEveryTeamEven": 0.315727, "pEveryTeamEvenFraction": "30492/96577", "lopsidedFrom": 8, "lopsidedLabel": "8-4 or worse", "pSomeTeamLopsided": 0.220347, "threeQuartersFrom": 9, "threeQuartersLabel": "9-3 or worse", "pSomeTeamThreeQuarters": 0.039126, "mostLopsidedTeam": [ { "largerGroup": 6, "label": "6-6", "exact": 0.315727 }, { "largerGroup": 7, "label": "7-5", "exact": 0.463926 }, { "largerGroup": 8, "label": "8-4", "exact": 0.181221 }, { "largerGroup": 9, "label": "9-3", "exact": 0.035797 }, { "largerGroup": 10, "label": "10-2", "exact": 0.003222 }, { "largerGroup": 11, "label": "11-1", "exact": 0.000107 }, { "largerGroup": 12, "label": "12-0", "exact": 0.000001 } ], "display": { "pEveryTeamEven": "31.6%", "pSomeTeamLopsided": "22.0%", "pSomeTeamThreeQuarters": "3.9%" } }, "3": { "teams": 3, "teamSize": 8, "pEveryTeamEven": 0.126842, "pEveryTeamEvenFraction": "12250/96577", "lopsidedFrom": 6, "lopsidedLabel": "6-2 or worse", "pSomeTeamLopsided": 0.386086, "threeQuartersFrom": 6, "threeQuartersLabel": "6-2 or worse", "pSomeTeamThreeQuarters": 0.386086, "mostLopsidedTeam": [ { "largerGroup": 4, "label": "4-4", "exact": 0.126842 }, { "largerGroup": 5, "label": "5-3", "exact": 0.487072 }, { "largerGroup": 6, "label": "6-2", "exact": 0.316597 }, { "largerGroup": 7, "label": "7-1", "exact": 0.065606 }, { "largerGroup": 8, "label": "8-0", "exact": 0.003883 } ], "display": { "pEveryTeamEven": "12.7%", "pSomeTeamLopsided": "38.6%", "pSomeTeamThreeQuarters": "38.6%" } }, "4": { "teams": 4, "teamSize": 6, "pEveryTeamEven": 0.059168, "pEveryTeamEvenFraction": "40000/676039", "lopsidedFrom": 5, "lopsidedLabel": "5-1 or worse", "pSomeTeamLopsided": 0.429119, "threeQuartersFrom": 5, "threeQuartersLabel": "5-1 or worse", "pSomeTeamThreeQuarters": 0.429119, "mostLopsidedTeam": [ { "largerGroup": 3, "label": "3-3", "exact": 0.059168 }, { "largerGroup": 4, "label": "4-2", "exact": 0.511712 }, { "largerGroup": 5, "label": "5-1", "exact": 0.378298 }, { "largerGroup": 6, "label": "6-0", "exact": 0.050822 } ], "display": { "pEveryTeamEven": "5.9%", "pSomeTeamLopsided": "42.9%", "pSomeTeamThreeQuarters": "42.9%" } }, "6": { "teams": 6, "teamSize": 4, "pEveryTeamEven": 0.017253, "pEveryTeamEvenFraction": "11664/676039", "lopsidedFrom": 4, "lopsidedLabel": "4-0 or worse", "pSomeTeamLopsided": 0.415679, "threeQuartersFrom": 3, "threeQuartersLabel": "3-1 or worse", "pSomeTeamThreeQuarters": 0.982747, "mostLopsidedTeam": [ { "largerGroup": 2, "label": "2-2", "exact": 0.017253 }, { "largerGroup": 3, "label": "3-1", "exact": 0.567068 }, { "largerGroup": 4, "label": "4-0", "exact": 0.415679 } ], "display": { "pEveryTeamEven": "1.7%", "pSomeTeamLopsided": "41.6%", "pSomeTeamThreeQuarters": "98.3%" } } } }, "friends": { "note": "Disjoint friend pairs; 'together' = both on the same team. P(one given pair together) = (s-1)/(n-1); expected pairs together = pairs * (s-1)/(n-1); the full distribution by inclusion–exclusion (see the script). When the two display strings would print the same (6 teams: about half), both get two decimals.", "pairs": 5, "byTeams": { "2": { "teams": 2, "teamSize": 12, "pSpecificPairTogether": 0.478261, "pSpecificPairTogetherFraction": "11/23", "expectedPairsTogether": 2.3913, "pNoPairTogether": 0.040613, "pAtLeastOnePairTogether": 0.959387, "distribution": [ { "pairsTogether": 0, "exact": 0.040613 }, { "pairsTogether": 1, "exact": 0.177682 }, { "pairsTogether": 2, "exact": 0.32152 }, { "pairsTogether": 3, "exact": 0.296137 }, { "pairsTogether": 4, "exact": 0.138069 }, { "pairsTogether": 5, "exact": 0.025979 } ], "byNumberOfPairs": [ { "pairs": 1, "pAtLeastOneTogether": 0.478261, "expectedTogether": 0.4783 }, { "pairs": 2, "pAtLeastOneTogether": 0.726708, "expectedTogether": 0.9565 }, { "pairs": 3, "pAtLeastOneTogether": 0.856162, "expectedTogether": 1.4348 }, { "pairs": 4, "pAtLeastOneTogether": 0.923851, "expectedTogether": 1.913 }, { "pairs": 5, "pAtLeastOneTogether": 0.959387, "expectedTogether": 2.3913 }, { "pairs": 6, "pAtLeastOneTogether": 0.978131, "expectedTogether": 2.8696 }, { "pairs": 8, "pAtLeastOneTogether": 0.993373, "expectedTogether": 3.8261 }, { "pairs": 10, "pAtLeastOneTogether": 0.997728, "expectedTogether": 4.7826 }, { "pairs": 12, "pAtLeastOneTogether": 0.998485, "expectedTogether": 5.7391 } ], "display": { "pAtLeastOnePairTogether": "95.9%", "pNoPairTogether": "4.1%" } }, "3": { "teams": 3, "teamSize": 8, "pSpecificPairTogether": 0.304348, "pSpecificPairTogetherFraction": "7/23", "expectedPairsTogether": 1.5217, "pNoPairTogether": 0.166293, "pAtLeastOnePairTogether": 0.833707, "distribution": [ { "pairsTogether": 0, "exact": 0.166293 }, { "pairsTogether": 1, "exact": 0.353602 }, { "pairsTogether": 2, "exact": 0.308831 }, { "pairsTogether": 3, "exact": 0.137447 }, { "pairsTogether": 4, "exact": 0.031001 }, { "pairsTogether": 5, "exact": 0.002827 } ], "byNumberOfPairs": [ { "pairs": 1, "pAtLeastOneTogether": 0.304348, "expectedTogether": 0.3043 }, { "pairs": 2, "pAtLeastOneTogether": 0.515152, "expectedTogether": 0.6087 }, { "pairs": 3, "pAtLeastOneTogether": 0.661383, "expectedTogether": 0.913 }, { "pairs": 4, "pAtLeastOneTogether": 0.762986, "expectedTogether": 1.2174 }, { "pairs": 5, "pAtLeastOneTogether": 0.833707, "expectedTogether": 1.5217 }, { "pairs": 6, "pAtLeastOneTogether": 0.883026, "expectedTogether": 1.8261 }, { "pairs": 8, "pAtLeastOneTogether": 0.941642, "expectedTogether": 2.4348 }, { "pairs": 10, "pAtLeastOneTogether": 0.970557, "expectedTogether": 3.0435 }, { "pairs": 12, "pAtLeastOneTogether": 0.985006, "expectedTogether": 3.6522 } ], "display": { "pAtLeastOnePairTogether": "83.4%", "pNoPairTogether": "16.6%" } }, "4": { "teams": 4, "teamSize": 6, "pSpecificPairTogether": 0.217391, "pSpecificPairTogetherFraction": "5/23", "expectedPairsTogether": 1.087, "pNoPairTogether": 0.29727, "pAtLeastOnePairTogether": 0.70273, "distribution": [ { "pairsTogether": 0, "exact": 0.29727 }, { "pairsTogether": 1, "exact": 0.403228 }, { "pairsTogether": 2, "exact": 0.225232 }, { "pairsTogether": 3, "exact": 0.064367 }, { "pairsTogether": 4, "exact": 0.009353 }, { "pairsTogether": 5, "exact": 0.000551 } ], "byNumberOfPairs": [ { "pairs": 1, "pAtLeastOneTogether": 0.217391, "expectedTogether": 0.2174 }, { "pairs": 2, "pAtLeastOneTogether": 0.386787, "expectedTogether": 0.4348 }, { "pairs": 3, "pAtLeastOneTogether": 0.518916, "expectedTogether": 0.6522 }, { "pairs": 4, "pAtLeastOneTogether": 0.622085, "expectedTogether": 0.8696 }, { "pairs": 5, "pAtLeastOneTogether": 0.70273, "expectedTogether": 1.087 }, { "pairs": 6, "pAtLeastOneTogether": 0.765844, "expectedTogether": 1.3043 }, { "pairs": 8, "pAtLeastOneTogether": 0.854096, "expectedTogether": 1.7391 }, { "pairs": 10, "pAtLeastOneTogether": 0.908556, "expectedTogether": 2.1739 }, { "pairs": 12, "pAtLeastOneTogether": 0.942348, "expectedTogether": 2.6087 } ], "display": { "pAtLeastOnePairTogether": "70.3%", "pNoPairTogether": "29.7%" } }, "6": { "teams": 6, "teamSize": 4, "pSpecificPairTogether": 0.130435, "pSpecificPairTogetherFraction": "3/23", "expectedPairsTogether": 0.6522, "pNoPairTogether": 0.500435, "pAtLeastOnePairTogether": 0.499565, "distribution": [ { "pairsTogether": 0, "exact": 0.500435 }, { "pairsTogether": 1, "exact": 0.367797 }, { "pairsTogether": 2, "exact": 0.112469 }, { "pairsTogether": 3, "exact": 0.017803 }, { "pairsTogether": 4, "exact": 0.001447 }, { "pairsTogether": 5, "exact": 0.000048 } ], "byNumberOfPairs": [ { "pairs": 1, "pAtLeastOneTogether": 0.130435, "expectedTogether": 0.1304 }, { "pairs": 2, "pAtLeastOneTogether": 0.243365, "expectedTogether": 0.2609 }, { "pairs": 3, "pAtLeastOneTogether": 0.341199, "expectedTogether": 0.3913 }, { "pairs": 4, "pAtLeastOneTogether": 0.426005, "expectedTogether": 0.5217 }, { "pairs": 5, "pAtLeastOneTogether": 0.499565, "expectedTogether": 0.6522 }, { "pairs": 6, "pAtLeastOneTogether": 0.563409, "expectedTogether": 0.7826 }, { "pairs": 8, "pAtLeastOneTogether": 0.667044, "expectedTogether": 1.0435 }, { "pairs": 10, "pAtLeastOneTogether": 0.745412, "expectedTogether": 1.3043 }, { "pairs": 12, "pAtLeastOneTogether": 0.804829, "expectedTogether": 1.5652 } ], "display": { "pAtLeastOnePairTogether": "49.96%", "pNoPairTogether": "50.04%" } } } }, "snakeDraft": { "note": "Deterministic comparison, not something the tool does: captains pick from a list sorted strongest first, in order 1..k then k..1. Ties in rating do not change any team total. CAVEAT: the gap is 0 here only because every team count studied gives an EVEN number of draft rounds (12, 8, 6, 4: each pair of rounds hands every team the same total) and because skill is known exactly and scored on the same scale used to draft. With an odd number of rounds the snake is not perfect (oddRoundsExample: 24 ranks into 8 teams of 3), and with real, uncertain skill it is only as good as the ranking. Never write that a snake draft always gives perfectly even teams.", "byTeams": { "2": { "rounds": 12, "ratingsGap": 0, "ranksGap": 0 }, "3": { "rounds": 8, "ratingsGap": 0, "ranksGap": 0 }, "4": { "rounds": 6, "ratingsGap": 0, "ranksGap": 0 }, "6": { "rounds": 4, "ratingsGap": 0, "ranksGap": 0 } }, "oddRoundsExample": { "teams": 8, "teamSize": 3, "rounds": 3, "ratingsGap": 2, "ranksGap": 7 } }, "fixesFourTeams": { "note": "4 teams of 6, rating mix 3/5/8/5/3. All values exact. 'close' = gap <= 3 (averages within 0.5); 'lopsided' = gap >= 6 (averages a full point apart). display uses one decimal in every row. 'Re-generate until close' is close and not lopsided BY CONSTRUCTION (you stop at the first close split); its meanGap is the expected gap of the split you keep, and the cost is meanPresses presses. The snake draft's 0 holds only for these even-round, exactly-known-skill conditions (see snakeDraft.note).", "rows": [ { "method": "One random split", "pClose": 0.14605, "pLopsided": 0.582007, "meanGap": 6.1592, "display": { "pClose": "14.6%", "pLopsided": "58.2%", "meanGap": "6.16" } }, { "method": "Random split + 1 hand swap", "pClose": 0.771035, "pLopsided": 0.04278, "meanGap": 2.5711, "display": { "pClose": "77.1%", "pLopsided": "4.3%", "meanGap": "2.57" } }, { "method": "Random split + 2 hand swaps", "pClose": 0.993261, "pLopsided": 0.0001, "meanGap": 0.8356, "display": { "pClose": "99.3%", "pLopsided": "<0.1%", "meanGap": "0.84" } }, { "method": "Tiered dealing (3 presses)", "pClose": 0.938776, "pLopsided": 0, "meanGap": 2.1122, "display": { "pClose": "93.9%", "pLopsided": "0%", "meanGap": "2.11" } }, { "method": "Re-generate until close", "pClose": 1, "pLopsided": 0, "meanGap": 2.4598, "byConstruction": true, "meanPresses": 6.847, "display": { "pClose": "100%", "pLopsided": "0%", "meanGap": "2.46" } }, { "method": "Snake draft (not random)", "pClose": 1, "pLopsided": 0, "meanGap": 0, "deterministic": true, "display": { "pClose": "100%", "pLopsided": "0%", "meanGap": "0.00" } } ] } }, "results": { "ratings": { "byTeams": { "2": { "splits": 400000, "pClose": 0.754907, "pLopsided": 0.062685, "pPerfectlyEven": 0.130713, "meanGap": 4.7508, "medianGap": 4, "largestGapSeen": 22, "distribution": [ { "gap": 0, "simulated": 0.130713 }, { "gap": 2, "simulated": 0.248968 }, { "gap": 4, "simulated": 0.211815 }, { "gap": 6, "simulated": 0.163412 }, { "gap": 8, "simulated": 0.113437 }, { "gap": 10, "simulated": 0.06897 }, { "gap": 12, "simulated": 0.036593 }, { "gap": 14, "simulated": 0.017305 }, { "gap": 16, "simulated": 0.006367 }, { "gap": 18, "simulated": 0.001932 }, { "gap": 20, "simulated": 0.000438 }, { "gap": 22, "simulated": 0.00005 } ] }, "3": { "splits": 200000, "pClose": 0.34307, "pLopsided": 0.286565, "pPerfectlyEven": 0.02188, "meanGap": 5.8574, 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"name": "swaps, ratings, 4 teams, 1 swap(s): mean gap", "kind": "simulated vs exact", "simulated": 2.57315625, "exact": 2.57110408, "se": 0.00166395, "z": 1.23, "tolerance": 0.00665578, "pass": true }, { "name": "swaps, ratings, 4 teams, 2 swap(s): P(close)", "kind": "simulated vs exact", "simulated": 0.9933575, "exact": 0.99326131, "se": 0.0000914691, "z": 1.05, "tolerance": 0.000365876, "pass": true }, { "name": "swaps, ratings, 4 teams, 2 swap(s): P(lopsided)", "kind": "simulated vs exact", "simulated": 0.0000925, "exact": 0.00010041, "se": 0.0000112028, "z": -0.71, "tolerance": 0.0000448112, "pass": true }, { "name": "swaps, ratings, 4 teams, 2 swap(s): mean gap", "kind": "simulated vs exact", "simulated": 0.837575, "exact": 0.83561856, "se": 0.00115971, "z": 1.69, "tolerance": 0.00463884, "pass": true }, { "name": "re-generate until 4-team gap <= 3: mean presses", "kind": "simulated vs exact", "simulated": 6.85615, "exact": 6.84698032, "se": 0.0316363, "z": 0.29, "tolerance": 0.126545, "pass": true }, { "name": "re-generate until 4-team gap <= 3: P(done within 3 presses)", "kind": "simulated vs exact", "simulated": 0.37805, "exact": 0.37727306, "se": 0.00242352, "z": 0.32, "tolerance": 0.00969408, "pass": true }, { "name": "re-generate until 4-team gap <= 3: P(more than 10 presses)", "kind": "simulated vs exact", "simulated": 0.209025, "exact": 0.2062175, "se": 0.00202294, "z": 1.39, "tolerance": 0.00809177, "pass": true }, { "name": "re-generate until 4-team gap <= 3: mean gap of the split kept", "kind": "simulated vs exact", "simulated": 2.456975, "exact": 2.45977813, "se": 0.00324276, "z": -0.86, "tolerance": 0.0129711, "pass": true }, { "name": "4 teams: P(gap <= 3 AND top 4 on different teams) per press", "kind": "simulated vs exact", "simulated": 0.03467, "exact": 0.03460108, "se": 0.00020434, "z": 0.34, "tolerance": 0.00081736, "pass": true }, { "name": "top players, 2 teams: P(two best on the same team) = 11/23", "kind": "simulated vs exact", 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0.00212999, "pass": true }, { "name": "top players, 6 teams: P(top 4 as evenly spread as possible)", "kind": "simulated vs exact", "simulated": 0.36242, "exact": 0.36137775, "se": 0.000759579, "z": 1.37, "tolerance": 0.00303831, "pass": true }, { "name": "top players, 6 teams: P(some team gets 3+ of the top 4)", "kind": "simulated vs exact", "simulated": 0.0458125, "exact": 0.04573687, "se": 0.000330322, "z": 0.23, "tolerance": 0.00132129, "pass": true }, { "name": "top players, 6 teams: P(top 8 as evenly spread as possible)", "kind": "simulated vs exact", "simulated": 0.18859, "exact": 0.18796118, "se": 0.000617721, "z": 1.02, "tolerance": 0.00247088, "pass": true }, { "name": "top players, 6 teams: P(some team gets none of the top 8)", "kind": "simulated vs exact", "simulated": 0.7780325, "exact": 0.77862349, "se": 0.000656447, "z": -0.9, "tolerance": 0.00262579, "pass": true }, { "name": "top players, 6 teams: P(some team gets 3+ of the top 8)", "kind": "simulated vs exact", 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