Parlay math: why four coin flips pay 13-to-1, and what an insider does to that number

A parlay is a bet that several outcomes all happen. The payout multiplies, which is why it feels like a lottery ticket, and the pricing is simple enough to put in one function.

American odds of −110 mean you risk 110 to win 100: a winning ticket returns 210/110 ≈ 1.909 × the stake. A four-leg parlay of −110 legs returns 1.909⁴ ≈ 13.3× — close to the $80,000 → $1.1 million (13.75×) ticket in the Jontay Porter case.

def decimal(american: int) -> float:
    return 1 + (american / 100 if american > 0 else 100 / -american)

def parlay(legs_american, p_each):
    """payout multiple and true win probability for independent legs"""
    mult = 1.0
    for a in legs_american:
        mult *= decimal(a)
    p = 1.0
    for q in p_each:
        p *= q
    return mult, p, mult * p   # multiple, hit rate, expected return per $1

print(parlay([-110]*4, [0.5]*4))          # (13.28, 0.0625, 0.83)
print(parlay([-110]*4, [0.9999]*4))       # (13.28, 0.9996, 13.28)

Two things to read off the numbers.

First, the honest case. Four genuine coin-flips at −110 hit 6.25% of the time and return about 83 cents per dollar: the sportsbook's ~4.5% edge per leg compounds to ~17% over the parlay. Parlays are expensive because the vig is charged four times, not because the multiplier is dishonest.

Second, the dishonest case. The multiplier does not change when one participant can make the legs certain. If a player can sit after three minutes, every "under" on his card resolves in his favour, each leg's probability goes to ~1, and the same 13.3× ticket returns ~13 dollars per dollar. The sportsbook's price assumed a coin; the insider was betting on a chair.

That asymmetry is specific to unders. An over needs minutes, the ball and made shots — nobody can guarantee it. An under needs only the player to stop. Which is why the NBA's fix in October 2026 was not a better line but no line: single-player under parlays, unders on low-minute players, opening-tip and missed-shot bets are gone from its sportsbook partners.

A general rule falls out of it: a market can be priced only when every participant watches the outcome from outside. Provably fair casino games get that property by construction (the result is a committed function nobody can touch after the bet); sports can only get it by withdrawing the bets an insider can decide.

The full write-up, with the NBA's findings and the court record: https://betkyo.com/en/blog/jontay-porter-the-under-only-the-player-can-guarantee-and-the-bets-the-nba-just-removed/