FAILURE MAP
← Case archive

FA-6346 / Statistics / Open access

Probability union with overlap · case 01

The overlapping event mass is counted twice.

Verified by executionVariant 1 · 4 checks per implementationDownload source bundle ↓JSON ↗

ROOT CAUSE

The overlapping event mass is counted twice.

VERIFIED REPAIR

Apply the specified mathematical contract directly, preserving all terms and boundary cases: return str(Fraction(a)+Fraction(b)-Fraction(overlap))

Unsuccessful approach: Independence is assumed despite an explicit overlap probability.

Case contract

Integer finite observations and equal lengths for paired samples. Counts and weights are nonnegative. Rational results use reduced Fraction strings. Empty or undefined statistics return None where shown. a,b,overlap are rational strings for valid probabilities with max(0,a+b-1)<=overlap<=min(a,b); do not assume independence. Exact operational definition: str(Fraction(a)+Fraction(b)-Fraction(overlap))

Why this case matters

Small exact fixtures expose this error without platform timing, external services, or probabilistic observations. Statistical estimators results depend on the stated convention.

1 / The failure

Exit 1
"""Failure Map reference implementation. Python standard library only."""
import json
import math
import calendar
import statistics
import itertools
from fractions import Fraction
from datetime import date, datetime, timedelta, timezone
from decimal import Decimal, ROUND_HALF_UP, ROUND_DOWN, ROUND_CEILING, ROUND_FLOOR

N = 1
observations = []
def solve(a, b, overlap):
    return str(Fraction(a)+Fraction(b))
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
check("fixture 1: ('1/2', '1/2', '1/4')", solve(*('1/2', '1/2', '1/4')), '3/4')
check("fixture 2: ('1/2', '1/2', '1/2')", solve(*('1/2', '1/2', '1/2')), '1/2')
check("fixture 3: ('0', '1/3', '0')", solve(*('0', '1/3', '0')), '1/3')
check("fixture 4: ('1/3', '1/3', '0')", solve(*('1/3', '1/3', '0')), '2/3')
print(json.dumps({"observations": observations, "passed": all(x["passed"] for x in observations)}, ensure_ascii=False))
raise SystemExit(0 if all(x["passed"] for x in observations) else 1)
Boundary fixtureActualExpectedOutcome
fixture 1: ('1/2', '1/2', '1/4')13/4Failed
fixture 2: ('1/2', '1/2', '1/2')11/2Failed
fixture 3: ('0', '1/3', '0')1/31/3Passed
fixture 4: ('1/3', '1/3', '0')2/32/3Passed

SHA-256 / 159265645540e075ebbbf7d357c8f46d13fb0ee3e244399bf4ff26ef49f8b47c

2 / The unsuccessful fix

Exit 1
"""Failure Map reference implementation. Python standard library only."""
import json
import math
import calendar
import statistics
import itertools
from fractions import Fraction
from datetime import date, datetime, timedelta, timezone
from decimal import Decimal, ROUND_HALF_UP, ROUND_DOWN, ROUND_CEILING, ROUND_FLOOR

N = 1
observations = []
def solve(a, b, overlap):
    return str(Fraction(a)+Fraction(b)-Fraction(a)*Fraction(b))
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
check("fixture 1: ('1/2', '1/2', '1/4')", solve(*('1/2', '1/2', '1/4')), '3/4')
check("fixture 2: ('1/2', '1/2', '1/2')", solve(*('1/2', '1/2', '1/2')), '1/2')
check("fixture 3: ('0', '1/3', '0')", solve(*('0', '1/3', '0')), '1/3')
check("fixture 4: ('1/3', '1/3', '0')", solve(*('1/3', '1/3', '0')), '2/3')
print(json.dumps({"observations": observations, "passed": all(x["passed"] for x in observations)}, ensure_ascii=False))
raise SystemExit(0 if all(x["passed"] for x in observations) else 1)
Boundary fixtureActualExpectedOutcome
fixture 1: ('1/2', '1/2', '1/4')3/43/4Passed
fixture 2: ('1/2', '1/2', '1/2')3/41/2Failed
fixture 3: ('0', '1/3', '0')1/31/3Passed
fixture 4: ('1/3', '1/3', '0')5/92/3Failed

SHA-256 / 4469b751b8bd570c92b675511d169b4b846657b5c858ec84572eee1d0bd97745

3 / The verified repair

Exit 0
"""Failure Map reference implementation. Python standard library only."""
import json
import math
import calendar
import statistics
import itertools
from fractions import Fraction
from datetime import date, datetime, timedelta, timezone
from decimal import Decimal, ROUND_HALF_UP, ROUND_DOWN, ROUND_CEILING, ROUND_FLOOR

N = 1
observations = []
def solve(a, b, overlap):
    return str(Fraction(a)+Fraction(b)-Fraction(overlap))
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
check("fixture 1: ('1/2', '1/2', '1/4')", solve(*('1/2', '1/2', '1/4')), '3/4')
check("fixture 2: ('1/2', '1/2', '1/2')", solve(*('1/2', '1/2', '1/2')), '1/2')
check("fixture 3: ('0', '1/3', '0')", solve(*('0', '1/3', '0')), '1/3')
check("fixture 4: ('1/3', '1/3', '0')", solve(*('1/3', '1/3', '0')), '2/3')
print(json.dumps({"observations": observations, "passed": all(x["passed"] for x in observations)}, ensure_ascii=False))
raise SystemExit(0 if all(x["passed"] for x in observations) else 1)
Boundary fixtureActualExpectedOutcome
fixture 1: ('1/2', '1/2', '1/4')3/43/4Passed
fixture 2: ('1/2', '1/2', '1/2')1/21/2Passed
fixture 3: ('0', '1/3', '0')1/31/3Passed
fixture 4: ('1/3', '1/3', '0')2/32/3Passed

SHA-256 / 4d209902992a5e3544817eebb8e6d7bf9e46883351394c7034cdd745cc3e05ab

Verification & scope

This reproducer isolates one failure mechanism. Results cover the supplied fixtures. Variants within a family share a test contract and should remain grouped when constructing evaluation splits. Related mechanisms with a shared evaluation_group must also remain together; these controlled models are not independent production incidents.

Observations recorded using Python 3.12.14 at 2026-09-29T14:38:00.596790+00:00.

Case digest / 16c87ce3368dfe42d0dbf2a1b0a02cff07297a2e34d32cb0cbd8083339b3a06d