The Outcome Tree
Every leg doubles the branches: n legs → 2n outcomes, but only the single golden path pays. Add legs and watch the tree explode — and your win probability shrink.
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8
total outcomes
12.5%
chance all legs hit
—
payout on $100
—
fair payout
Expected value of a $100 parlay
—
The math, worked out
Take a 3-leg parlay at standard −110 odds per leg (bet $110 to win $100):
- Decimal odds per leg: −110 → 1.909. Parlay payout multiplies: 1.909³ = 6.96× your stake back.
- True probability (coin-flip legs): 0.5³ = 12.5%, so a fair payout would be 1/0.125 = 8×.
- Expected value: 0.125 × $696 − $100 = −$13 per $100. The single-leg edge (~4.5%) compounds to ~13% over 3 legs, and ~31% over 8 legs.
"But what if I cover every outcome?"
The tweet's dream app would generate all permutations (round-robin betting does something like this — betting every 2-leg combo out of 6 picks is C(6,2) = 15 separate parlays). But here's the theorem-level problem: expected values add. If every individual bet has negative EV, any weighted combination of them — including "cover everything" — also has negative EV. Covering all 2n outcomes at book odds guarantees a loss equal to the total vig, roughly:
- Bet $1 on all 8 outcomes of a 3-leg coin-flip parlay: total staked $8, guaranteed return 6.96 → guaranteed loss of ~$1.04 (13%), every time.
- The only historical exceptions are pricing mistakes (arbitrage), which books close within minutes and ban accounts for.
| Legs | Outcomes | Win chance (50% legs) | House edge at −110 |
|---|---|---|---|
| 2 | 4 | 25% | ~8.9% |
| 4 | 16 | 6.25% | ~17% |
| 6 | 64 | 1.56% | ~24% |
| 8 | 256 | 0.39% | ~31% |
The real "knowledge of statistics that can help bettors" is this: parlays are the highest-margin product a sportsbook sells. If you gamble, treat it as paid entertainment with a budget — and if it stops feeling optional, help exists at 1-800-GAMBLER (US).