from __future__ import annotations

import math


def _norm_cdf(z: float) -> float:
    # Standard normal CDF via erf (no scipy dependency)
    return 0.5 * (1.0 + math.erf(z / math.sqrt(2.0)))


def fair_prob_threshold(
    *,
    spot: float,
    strike: float,
    sigma_per_second: float,
    time_remaining_seconds: float,
    resolves_yes_if_spot_ge_strike: bool,
) -> float:
    """
    Conservative approximation:
      spot(t) ~ Normal(spot, spot*sigma*sqrt(t))
    and compute P(spot_T >= strike) or P(spot_T <= strike).
    """
    if spot <= 0 or strike <= 0:
        return 0.5
    t = max(1.0, float(time_remaining_seconds))
    sigma = max(1e-9, float(sigma_per_second))
    stdev = spot * sigma * math.sqrt(t)
    if stdev <= 0:
        return 0.5

    z = (spot - strike) / stdev
    p_ge = _norm_cdf(z)  # P(spot_T >= strike)

    fair = p_ge if resolves_yes_if_spot_ge_strike else (1.0 - p_ge)

    # clip away from certainty (tail risk)
    return max(0.03, min(0.97, fair))