{
  "kind": "invented_independent_bernoulli_shots_not_match_prediction",
  "calculated_on": "2026-11-02",
  "cases": [
    {
      "id": "one_half",
      "probabilities": [
        "0.5"
      ],
      "shots": 1,
      "expected_goals": "1/2",
      "no_goal": "1/2",
      "at_least_one": "1/2",
      "distribution": [
        "1/2",
        "1/2"
      ]
    },
    {
      "id": "five_tenths",
      "probabilities": [
        "0.1",
        "0.1",
        "0.1",
        "0.1",
        "0.1"
      ],
      "shots": 5,
      "expected_goals": "1/2",
      "no_goal": "59049/100000",
      "at_least_one": "40951/100000",
      "distribution": [
        "59049/100000",
        "6561/20000",
        "729/10000",
        "81/10000",
        "9/20000",
        "1/100000"
      ]
    },
    {
      "id": "two_halves",
      "probabilities": [
        "0.5",
        "0.5"
      ],
      "shots": 2,
      "expected_goals": "1",
      "no_goal": "1/4",
      "at_least_one": "3/4",
      "distribution": [
        "1/4",
        "1/2",
        "1/4"
      ]
    },
    {
      "id": "ten_tenths",
      "probabilities": [
        "0.1",
        "0.1",
        "0.1",
        "0.1",
        "0.1",
        "0.1",
        "0.1",
        "0.1",
        "0.1",
        "0.1"
      ],
      "shots": 10,
      "expected_goals": "1",
      "no_goal": "3486784401/10000000000",
      "at_least_one": "6513215599/10000000000",
      "distribution": [
        "3486784401/10000000000",
        "387420489/1000000000",
        "387420489/2000000000",
        "14348907/250000000",
        "11160261/1000000000",
        "3720087/2500000000",
        "137781/1000000000",
        "2187/250000000",
        "729/2000000000",
        "9/1000000000",
        "1/10000000000"
      ]
    },
    {
      "id": "certain_zero",
      "probabilities": [
        "0",
        "0"
      ],
      "shots": 2,
      "expected_goals": "0",
      "no_goal": "1",
      "at_least_one": "0",
      "distribution": [
        "1",
        "0",
        "0"
      ]
    },
    {
      "id": "certain_one",
      "probabilities": [
        "0",
        "1"
      ],
      "shots": 2,
      "expected_goals": "1",
      "no_goal": "0",
      "at_least_one": "1",
      "distribution": [
        "0",
        "1",
        "0"
      ]
    }
  ],
  "method": "IndependentBernoulli assumptions; exactFraction dynamicprogramming crosschecked exhaustive2^Noutcomes. Expectedgoals sum doesnotrequireindependence; productanddistribution here do.",
  "scope": "Probabilities are chosen educational inputs, not output from a fitted xG model. Actual shots/rebounds may be dependent and future shot opportunities change with game state. Notmatchobservation, scoreforecast, playerability rating or betting tool."
}