{
  "title": "Finding patterns by chance",
  "data_kind": "Exact enumeration of all fair independent 16-bit strings; no measured or decoded messages",
  "bits_per_string": 16,
  "word_length": 4,
  "equally_likely_strings": 65536,
  "window_starts": "first w starts, counted from 0 through w-1; w=1..13",
  "pattern_order": [
    "1010",
    "0000",
    "0001",
    "0010",
    "0011",
    "0100",
    "0101",
    "0110",
    "0111",
    "1000",
    "1001",
    "1011",
    "1100",
    "1101",
    "1110",
    "1111"
  ],
  "pattern_selection": "first k entries of the prespecified order, k=1..16",
  "event": "At least one of the k allowed 4-bit words appears at one of the w tested positions.",
  "exact_formula": "P(match)=count_of_matching_16_bit_strings/65536",
  "one_position_formula": "P(match)=k/16",
  "union_bound": "P(match)<=min(1,w*k/16)",
  "warning": "1-(1-k/16)^w assumes independent windows and is generally not exact for overlapping windows. Different word sets can have different overlap behaviour.",
  "validation": {
    "one_position_closed_form": "passed for all k",
    "single_word_avoidance_automaton_matching_count": 35789,
    "nonoverlapping_four_window_matching_count": 14911,
    "nonoverlapping_formula": "1-(15/16)^4",
    "monotonicity": "passed",
    "union_bound": "passed"
  },
  "limitations": [
    "Each bit is independent and equally likely to be zero or one.",
    "Search choices are fixed before enumeration; a real analyst may try other word lengths, offsets, transforms or codebooks too.",
    "A match is a combinatorial coincidence, not evidence of an intended message.",
    "Results depend on word length, string length and which patterns are allowed.",
    "No uncertainty intervals: all strings in this finite null model are enumerated exactly."
  ]
}