Quantifying ATS Winning & Losing Streaks

The Result significance section on the query pages shows you a system's max win streak and max losing streak directly. This article is the "how would a statistician actually check whether that streak means anything" companion — five named, real methods, with the formulas written out, that quantify streak behavior beyond just eyeballing the number.

Setup: a streak is a sequence problem, not just a record

A won-loss record (say, 58-42 ATS) throws away information: it tells you the total, but not the order. Two systems can have the identical 58-42 record while one alternates W/L almost every game and the other goes on a 14-game win streak followed by a 9-game losing streak. The methods below all use the order — treating each decided game as one term in a chronological sequence of W's and L's, exactly the sequence the site's own max_win_streak/max_losing_streak calculation walks through game by game.

1. The Wald–Wolfowitz Runs Test

This is the direct test for "is this sequence actually random, or is there real streakiness (or anti-streakiness) in it?" A run is a maximal block of identical results — the sequence W W L L L W L W W W has 5 runs: WW, LLL, W, L, WWW. Under the null hypothesis that wins and losses are independent and identically distributed (i.e. no real streak tendency, just a fixed win rate applied randomly game to game), the number of runs has a known distribution.

Let n₁ = number of wins, n₂ = number of losses, n = n₁ + n₂, and R = the observed number of runs.

Expected runs:   μ_R = 1 + (2 × n₁ × n₂) / n

Variance:   σ²_R = [2 × n₁ × n₂ × (2 × n₁ × n₂ − n)] / [n² × (n − 1)]

Test statistic:   Z = (R − μ_R) / σ_R

Z is compared against the standard normal distribution (for reasonably-sized samples). A large negative Z means fewer runs than chance would produce — real streakiness, wins clumping with wins and losses with losses. A large positive Z means more runs than chance — the record alternates suspiciously often, sometimes a sign of anti-correlation (e.g. a market over-correcting after every result).

Worked example: a system goes 30-20 ATS (n₁=30, n₂=20, n=50) across R=18 observed runs.
μ_R = 1 + (2×30×20)/50 = 1 + 24 = 25
σ²_R = [2×30×20×(1200−50)] / [50²×49] = 1,380,000 / 122,500 ≈ 11.27 → σ_R ≈ 3.36
Z = (18 − 25) / 3.36 ≈ −2.08 — comfortably beyond the usual ±1.96 cutoff, so this record has meaningfully fewer runs (longer, clumpier streaks) than 50 independent coin-flip-style bets would typically produce. That's evidence of real streak behavior in the sequence itself — separate from, and complementary to, the system's own win-rate p-value (see the companion article on p-values).

2. Expected Longest Run (Schilling's Approximation)

This answers a different, very practical question: "given how often this system wins, how long a streak should I expect to see purely by chance over N games?" — the baseline your actual max_win_streak number should be compared against before calling it remarkable. For n independent trials each with success probability p, the expected length of the longest run of successes is given by an approximation formalized by Mark F. Schilling ("The Longest Run of Heads," The College Mathematics Journal, 1990), building on earlier work by Erdős and Rényi:

E[L_n] ≈ log_1/p(n × q) + γ / ln(1/p) − 1/2

where q = 1 − p, log_1/p(x) = ln(x) / ln(1/p), and γ ≈ 0.5772 is the Euler–Mascheroni constant. (A small oscillating correction term is omitted here — negligible for practical bankroll/streak planning.)

Worked example: a fair (p=0.5) ATS system over a 100-game sample.
log₂(100 × 0.5) = log₂(50) ≈ 5.64
γ/ln(2) ≈ 0.5772/0.6931 ≈ 0.833
E[L₁₀₀] ≈ 5.64 + 0.833 − 0.5 ≈ 5.97 games
In plain terms: a perfectly average, no-edge 50% system, run for 100 games, would be expected to produce a win streak of roughly 6 games just from chance. A system advertising a "10-game win streak!" over a similar sample isn't automatically special — Lesson 3 covers this exact trap (small-sample cherry-picking) in the main Learn library.

Schilling, M.F. (1990). "The Longest Run of Heads." The College Mathematics Journal, 21(3), 196–207.

3. The Geometric Distribution (one streak's own length)

Once a streak has started, how long should any single run of wins (or losses) last before it breaks? If each game is independent with win probability p, the length of one streak follows a geometric distribution:

P(streak length = k) = p^k−1 × (1 − p)

the probability a winning streak runs exactly k games before the first loss. Its expected value, E[k] = 1 / (1 − p), is the average length you'd expect ANY winning streak (not the longest one — see #2 above) to reach before ending.

Worked example: at p=0.55 (a real, modest edge), the expected length of any given winning streak is 1/(1−0.55) = 2.22 games — even a genuinely good system's streaks are short on average; it's the tail (the occasional long run, governed by #2's formula) that produces the eye-catching numbers.

4. Conditional Win Rate After a Win vs. After a Loss (the "Hot Hand" Test)

This is the formal version of the question everyone actually means when they ask "does this system get hot?" — is the win probability actually different immediately after a win versus immediately after a loss? This is the same test structure Gilovich, Vallone & Tversky used in their famous 1985 "hot hand" basketball study (Cognitive Psychology, 17(3), 295–314), applied here to ATS results instead of made/missed shots. It's a two-proportion z-test:

z = (p̂₁ − p̂₂) / √[ p̂(1−p̂)(1/n₁ + 1/n₂) ]

where p̂₁ = win rate on games immediately following a win (n₁ such games), p̂₂ = win rate immediately following a loss (n₂ such games), and p̂ is the pooled win rate across both groups combined.

This is conceptually the same question the site's own "Record after a loss" feature answers, from the opposite direction — that feature reports the actual record and its own significance test for "after a loss" specifically; this z-test is the textbook way to formally compare BOTH conditional rates (after-win and after-loss) against each other directly, rather than either one against a flat 50% baseline.

5. Maximum Drawdown (the dollar-based streak metric)

The four methods above all describe the win/loss sequence. Maximum drawdown, already on the query pages as its own tile, is the equivalent concept measured in real dollars against a flat-stake equity curve rather than in games:

MDD = max_{t₂ > t₁} [ Peak(t₁) − Equity(t₂) ]

the largest decline from any running peak to a later low point in the equity curve, tracked game by game in real chronological order — see the Portfolio Approach lesson for why this can differ from (and sometimes exceed) what the losing-streak count alone suggests.

Traders sometimes also compute the Calmar ratio (annualized return divided by maximum drawdown) as a single number combining profitability and downside risk — worth knowing the name if you ever see it elsewhere, though this site reports the two pieces (ROI and max drawdown) separately rather than pre-combined, so you can weigh them yourself.

The thread connecting all five methods: none of them tell you whether a streak will continue, reverse, or mean anything about tomorrow's game. They tell you whether what already happened is distinguishable from what pure chance alone would have produced at the same win rate and sample size. That distinction — describing the past rigorously vs. predicting the future — is the entire reason this article exists as a companion to the Portfolio Approach lesson's gambler's-fallacy warning, not a replacement for it.

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