Betting apps sell a feeling: the next bet could be the one. Statistics sells a different product: what happens over many bets. The two are not in conflict — short-run wins are real and frequent — but only one of them describes the bettor's bank balance after a year. This note builds that picture from first principles, with every number computed in a reproducible script.
§1Odds are prices
A decimal odd \(o\) pays \(o\) per unit staked if the bet wins (stake included). The probability that would make the bet fair is \(1/o\), the implied probability. In a fair market, the implied probabilities of all outcomes sum to 1. In a real one, they don't:
| outcome | offered odds | implied prob. 1/o | fair odds |
|---|---|---|---|
| home | 2.10 | 47.6% | 2.20 |
| draw | 3.40 | 29.4% | 3.56 |
| away | 3.60 | 27.8% | 3.77 |
| sum | 104.8% |
Illustrative football 1X2 market, not from a specific bookmaker. The sum exceeds 100%: the excess is the margin.
The excess over 100% — here 4.8% — is the overround, the bookmaker's margin. Dividing each implied probability by the sum removes it and gives the fair odds. Every price the bettor sees has been shaded down from its fair value.
§2Expected value: the loss is built into the price
Let \(p\) be the true probability that a bet wins. The gain \(G\) per unit staked is \(o - 1\) with probability \(p\) and \(-1\) otherwise:
If the market is efficient — its fair probabilities equal the true ones, \(p = q\) — and each price carries the margin \(m\), then \(o = 1/\big(q(1+m)\big)\) and
Two consequences. First, the expected loss does not depend on which outcome you pick: favourite, draw or underdog, each unit staked loses \(m/(1+m)\) on average — 4.6% in the market above. Second, nothing about the bettor's skill enters (2). To change the sign, you need \(p > 1/o\): a better probability than the market's, by more than its margin (§8).
§3The law of large numbers
Take the simplest reference bet: odds 1.91 on a genuine 50/50 event, a common price for “even” markets. Then \(\mu = \mathbb{E}[G] = 0.5 \times 1.91 - 1 = -4.5\%\) and the standard deviation of a single bet is \(\sigma = o\sqrt{p(1-p)} = 0.955\) — the noise is twenty times larger than the signal. That is why a single bet tells you almost nothing.
The (strong) law of large numbers says the average return per bet converges to the expectation:
Early on, the paths are everywhere — some bettors are up 20% or more. The band narrows like \(1/\sqrt n\), and all paths are pulled toward the same line. The law does not say losses are guaranteed on any given day; it says the average is not negotiable.
§4The central limit theorem: when luck runs out
The total result \(S_n = \sum G_i\) behaves differently from the average. Its expected value falls linearly, \(n\mu\), while the luck around it grows only like \(\sigma\sqrt n\). By the central limit theorem,
Linear beats square root. The expected loss overtakes two standard deviations of luck at
At five bets a day, that is about a year. The exact binomial probabilities:
| bets n | P(ahead) | expected loss (u) | ±2σ√n (u) |
|---|---|---|---|
| 1 | 50.00% | 0.0 | 1.9 |
| 10 | 37.70% | 0.5 | 6.0 |
| 50 | 33.59% | 2.3 | 13.5 |
| 100 | 30.86% | 4.5 | 19.1 |
| 500 | 15.18% | 22.5 | 42.7 |
| 1,000 | 6.86% | 45.0 | 60.4 |
| 2,000 | 1.68% | 90.0 | 85.4 |
| 5,000 | 0.04% | 225.0 | 135.1 |
| 10,000 | < 0.01% | 450.0 | 191.0 |
One-unit bets at odds 1.91 on a 50/50 event. P(ahead) computed exactly from the binomial. Once the expected loss exceeds 2σ√n, luck can no longer cover the margin.
After 100 bets, almost one bettor in three is still ahead — which is exactly what keeps the experience convincing. After 2,000, fewer than 2%. Try it:
§5Accumulators: the margin compounds
A multiple (parlay) multiplies the odds of \(k\) selections. It also multiplies the margins. With independent legs, each priced as in (2),
| legs k | expected value per unit staked |
|---|---|
| 1 | -4.6% |
| 2 | -9.0% |
| 3 | -13.1% |
| 5 | -20.9% |
| 8 | -31.3% |
| 10 | -37.5% |
Each leg priced with the margin of the market above (4.8%). The margin compounds.
A 5-leg accumulator gives back about 79 cents per real staked in expectation: the house edge is roughly four and a half times that of a single bet. Big multipliers on the screen are the visible side of the same arithmetic.
§6Gambler's ruin
Real bankrolls are finite. Suppose a bettor starts with \(k\) units, bets one unit at odds 2.00 with win probability \(p\), and stops on reaching \(N\) or zero. With \(r = (1-p)/p\), the classical result is
| win prob. p | edge per bet | P(double 50 → 100) | P(ruin) |
|---|---|---|---|
| 0.500 | 0.0% | 50.00% | 50.00% |
| 0.495 | -1.0% | 26.89% | 73.11% |
| 0.490 | -2.0% | 11.92% | 88.08% |
| 0.480 | -4.0% | 1.79% | 98.21% |
| 0.475 | -5.0% | 0.67% | 99.33% |
| 0.450 | -10.0% | < 0.01% | ≈ 100% |
One-unit bets at odds 2.00, starting with 50 units, stopping at 100 or at 0 (gambler's ruin).
In a fair game, the chance of doubling 50 units is 50%. A disadvantage of just two percentage points (\(p = 0.49\)) cuts it to 12%; four points cuts it to under 2%. Small edges become decisive because the walk is long.
§7Fallacies the numbers rule out
- “It's due.” Independent events have no memory: after five losses, \(p\) is still \(p\). The gambler's fallacy confuses the law of large numbers — which works by diluting past deviations — with a force that corrects them.
- Streaks. In 100 fair coin flips, the longest run is typically 6 or 7. Streaks are what randomness looks like, not evidence of form.
- Survivorship. From the table in §4, about 31% of bettors are ahead after 100 bets. Among a million players, that is hundreds of thousands of genuine winning stories — selected from a population that, in aggregate, loses 4.5% of everything it stakes.
- Regression to the mean. A great month is usually followed by a more ordinary one — not because luck “turned”, but because extreme results are partly noise.
§8Can anyone win? Kelly and the edge
Yes — if they consistently estimate \(p\) better than the market by more than its margin, so that \(p\,o - 1 > 0\). Even then, stake size matters. Betting a fraction \(f\) of the bankroll at net odds \(b = o - 1\), the long-run growth rate of capital is
\(f^{*}\) is the Kelly fraction. A bettor who wins 52% of bets at odds 2.10 — a genuine edge of \(p\,o - 1 = 9.2\%\) per bet — should stake about 8.4% per bet, for a growth of only 0.38% per bet. Betting twice that fraction drives the growth rate to about zero; more than that, and capital shrinks despite the edge. And when there is no edge, (8) gives \(f^{*} < 0\): the optimal stake is zero. In practice, bookmakers can also limit accounts that win consistently, which caps even the rare real edge.
§9Takeaways
- The margin is in the price: every unit staked loses \(m/(1+m)\) on average, whichever outcome is chosen.
- The average converges (LLN); the total diverges downward, because \(n|\mu|\) grows faster than \(\sigma\sqrt n\) (CLT). The crossover is around 1,800 bets for a 4.5% edge.
- Short-run wins are frequent and real — that is the product, not a flaw in the math.
- Accumulators compound the margin; finite bankrolls turn small edges into near-certain ruin.
- Winning requires better probabilities than the market and disciplined staking — and, absent an edge, the optimal bet is no bet.
Code: research/bets/bets.py (standard library, fixed seeds). References: J. L. Kelly Jr. (1956), A New Interpretation of Information Rate, Bell System Technical Journal; W. Feller, An Introduction to Probability Theory and Its Applications, vol. 1 (gambler's ruin); R. H. Thaler & W. T. Ziemba (1988), Anomalies: Parimutuel Betting Markets, Journal of Economic Perspectives.