{
  "schemaVersion": 1,
  "generatedAt": "2026-09-11T00:00:00Z",
  "title": "Pocket aces waiting-time distribution lab",
  "assumptions": [
    "Standard 52-card Texas Hold’em deck with two private cards for one fixed seat.",
    "Every hand is freshly and independently shuffled; no jokers, exposed cards, misdeals, or variant deck.",
    "A first-hit wait includes the successful hand. Wall-clock time is not modeled.",
    "The combined pocket-pair/flop event deals a hypothetical three-card runout every hand and asks whether a pocket pair is dealt and at least one of its two remaining rank-mates appears, regardless of whether live action would reveal a flop.",
    "Card frequency is not strategic EV, profitability, or evidence that one observed shuffle is fair or unfair."
  ],
  "exactCounts": {
    "totalTwoCardCombinations": "1326",
    "pocketAcesCombinations": "6",
    "conditionalFlopsAfterPocketPair": "19600",
    "conditionalFlopsWithRankMatch": "2304",
    "conditionalFlopRankMatchFraction": "144/1225",
    "conditionalFlopRankMatchProbability": 0.11755102040816326
  },
  "equations": {
    "atLeastOneByN": "1 - (1 - p)^n",
    "firstHitMean": "1 / p",
    "firstHitQuantile": "ceil(log(1 - q) / log(1 - p)), checked for the first integer meeting the CDF threshold",
    "repeatWait": "P(T_r <= n) = P(Binomial(n, p) >= r)"
  },
  "events": [
    {
      "id": "exact_suited_combo",
      "labelEn": "One exact suited combo (for example A♠K♠)",
      "labelJa": "特定のスーテッド1コンボ（例：A♠K♠）",
      "scopeEn": "One specified unordered two-card combination.",
      "scopeJa": "指定した2枚の組み合わせ1通り。",
      "numerator": "1",
      "denominator": "1326",
      "probability": 0.0007541478129713424,
      "percent": 0.07541478129713425,
      "oneIn": 1326,
      "meanHands": 1326,
      "medianHands": 919,
      "percentile90Hands": 3053,
      "percentile95Hands": 3971
    },
    {
      "id": "pocket_aces",
      "labelEn": "Pocket aces (AA)",
      "labelJa": "ポケットエース（AA）",
      "scopeEn": "Any two of the four aces.",
      "scopeJa": "4枚のAから2枚。",
      "numerator": "1",
      "denominator": "221",
      "probability": 0.004524886877828055,
      "percent": 0.4524886877828055,
      "oneIn": 221,
      "meanHands": 221,
      "medianHands": 153,
      "percentile90Hands": 508,
      "percentile95Hands": 661
    },
    {
      "id": "ace_king_any_suits",
      "labelEn": "Ace-king, any suits (AK)",
      "labelJa": "AK（スート不問）",
      "scopeEn": "Four aces multiplied by four kings.",
      "scopeJa": "Aが4枚、Kが4枚の16コンボ。",
      "numerator": "8",
      "denominator": "663",
      "probability": 0.012066365007541479,
      "percent": 1.206636500754148,
      "oneIn": 82.875,
      "meanHands": 82.875,
      "medianHands": 58,
      "percentile90Hands": 190,
      "percentile95Hands": 247
    },
    {
      "id": "queens_or_better",
      "labelEn": "QQ, KK or AA",
      "labelJa": "QQ・KK・AA",
      "scopeEn": "Three specified pocket-pair ranks.",
      "scopeJa": "指定したポケットペア3ランク。",
      "numerator": "3",
      "denominator": "221",
      "probability": 0.013574660633484163,
      "percent": 1.3574660633484164,
      "oneIn": 73.66666666666667,
      "meanHands": 73.66666666666667,
      "medianHands": 51,
      "percentile90Hands": 169,
      "percentile95Hands": 220
    },
    {
      "id": "any_pocket_pair",
      "labelEn": "Any pocket pair",
      "labelJa": "いずれかのポケットペア",
      "scopeEn": "Any of 13 ranks, six combinations per rank.",
      "scopeJa": "13ランク、それぞれ6コンボ。",
      "numerator": "1",
      "denominator": "17",
      "probability": 0.058823529411764705,
      "percent": 5.88235294117647,
      "oneIn": 17,
      "meanHands": 17,
      "medianHands": 12,
      "percentile90Hands": 38,
      "percentile95Hands": 50
    },
    {
      "id": "pocket_pair_flop_rank_match",
      "labelEn": "Pocket pair, then at least one rank-mate in a hypothetical flop runout",
      "labelJa": "ポケットペアの後、仮想フロップに同ランクが1枚以上",
      "scopeEn": "Deals a hypothetical three-card runout every hand, whether or not live action would reveal a flop. Includes a set, a full house on a paired flop, or quads; it is not an observed-flop rate or profitability test.",
      "scopeJa": "実戦でフロップが公開されるかに関係なく、毎ハンド仮想の3枚を配る。セット、ペアボードでのフルハウス、クワッズを含み、実際に見たフロップの比率や収益性ではない。",
      "numerator": "144",
      "denominator": "20825",
      "probability": 0.006914765906362545,
      "percent": 0.6914765906362544,
      "oneIn": 144.61805555555554,
      "meanHands": 144.61805555555554,
      "medianHands": 100,
      "percentile90Hands": 332,
      "percentile95Hands": 432
    }
  ],
  "pocketAcesDroughts": [
    {
      "hands": 50,
      "seenAtLeastOnce": 0.2028861468565733,
      "noPocketAces": 0.7971138531434268
    },
    {
      "hands": 100,
      "seenAtLeastOnce": 0.36460950512683954,
      "noPocketAces": 0.6353904948731605
    },
    {
      "hands": 153,
      "seenAtLeastOnce": 0.5003656461101654,
      "noPocketAces": 0.49963435388983457
    },
    {
      "hands": 221,
      "seenAtLeastOnce": 0.6329544380210141,
      "noPocketAces": 0.36704556197898586
    },
    {
      "hands": 500,
      "seenAtLeastOnce": 0.8964372513358122,
      "noPocketAces": 0.1035627486641878
    },
    {
      "hands": 508,
      "seenAtLeastOnce": 0.9001272920173669,
      "noPocketAces": 0.09987270798263315
    },
    {
      "hands": 661,
      "seenAtLeastOnce": 0.950100164075869,
      "noPocketAces": 0.04989983592413105
    },
    {
      "hands": 1000,
      "seenAtLeastOnce": 0.9892747570891183,
      "noPocketAces": 0.010725242910881727
    }
  ],
  "pocketAcesRepeatWaits": [
    {
      "targetOccurrences": 1,
      "meanHands": 221,
      "medianHands": 153,
      "percentile90Hands": 508,
      "percentile95Hands": 661
    },
    {
      "targetOccurrences": 5,
      "meanHands": 1105,
      "medianHands": 1032,
      "percentile90Hands": 1765,
      "percentile95Hands": 2021
    },
    {
      "targetOccurrences": 10,
      "meanHands": 2210,
      "medianHands": 2137,
      "percentile90Hands": 3137,
      "percentile95Hands": 3468
    },
    {
      "targetOccurrences": 20,
      "meanHands": 4420,
      "medianHands": 4347,
      "percentile90Hands": 5722,
      "percentile95Hands": 6157
    }
  ]
}