How wide is 'wide' when I see a 3-of-3, 4-of-4, or 5-of-5 unanimity?

Exact 95% one-sided upper CIs on the true rate: n=3 unanimity → 63.16%; n=4 → 52.71%; n=5 → 45.07%; n=6 → 39.30%; n=10 → 25.89%; n=15 → 18.10%; n=20 → 13.91%; n=30 → 9.51%. For n=3 the 'unanimity' allows the true rate anywhere up to almost 2-in-3. Only at n≥10 does unanimity start feeling meaningfully constrained (below 26%). At n=30 the unanimity is what most trading contexts would call 'sharp' — but n=30 tail-bucket samples are exceptionally rare on quarterly macro events where 15 years of data give at most 60 releases and the tail bucket holds 3-8 of them.

How does this compare to Wilson score, Clopper-Pearson, and Agresti-Coull for the x=0 corner case?

All four intervals treat x=0 correctly (unlike naive Wald which collapses to [0, 0]) but they disagree on width. For n=4, x=0 at 95%: naive Wald [0, 0] (useless); Wilson score [0, ~0.49]; Agresti-Coull [0, ~0.55]; Rule of Three ≈ [0, 0.75] (upper only); Clopper-Pearson exact two-sided [0, 0.6024]; one-sided exact (which the Rule of Three approximates) [0, 0.5271]. The one-sided exact = 1 - α^(1/n) is what Rule of Three approximates. Two-sided Clopper-Pearson uses α/2 = 0.025 and gives 1 - 0.025^(1/n) = 0.6024 for n=4. Wilson score is narrower because it doesn't require exact frequentist coverage. Rule of Three is the widest (safest) — good for the 'if I see 0 events, what's the WORST-CASE true rate' quick calculation.

Why should I care as a trader?

Every slot-1 tail-unanimity post on Vantage Insights carries an implicit small-sample caveat that Rule of Three makes concrete. If you see '4-of-4 big-miss down at 15m' and start trading on the assumption that Japan-GDP-big-miss → GBPJPY-down is a 100% reliable signal, Rule of Three tells you the true rate might be 47% (or 25%, or 5% — anywhere below 53%). Position size accordingly. This is the same lesson quality-control literature learned from Hanley & Lippman-Hand 1983: if a manufacturer runs 20 quality tests and gets 0 failures, the true failure rate could still be as high as 15% (or 0%; the sample can't distinguish). For quarterly-macro FX events with tail buckets of size 3-6, the CI is even wider.

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Statistics9 September 2026· 5 min read

Statistics for Traders #40: The Rule of Three — for n Bernoulli trials with 0 successes, the 95% upper confidence bound on the true success rate is approximately 3/n. Direct application to today's slot 1 (GBPJPY × Japan Prelim GDP: 0-of-4 UP at 15m) gives a 95% upper CI of 52.7% — the 4-of-4 unanimity is CONSISTENT WITH a true up-rate as high as 1-in-2. Opens a new small-sample-inference thread.

Q: What's the one-line summary?

For n Bernoulli trials with 0 successes, the one-sided 95% upper confidence bound on the true success probability is p_U = 1 - 0.05^(1/n), which is well-approximated by 3/n for n larger than about 10. Solved: (1-p_U)^n = 0.05 → p_U = 1 - 0.05^(1/n). Taylor: for large n, ln(1-p_U) ≈ -p_U so -n*p_U ≈ ln(0.05) ≈ -2.996 → p_U ≈ 2.996/n ≈ 3/n. Direct application to today's slot 1: GBPJPY × Japan Prelim GDP big-miss = 0-of-4 UP at 15m → exact 95% upper CI = 1 - 0.05^(1/4) = 52.71% → the 4-of-4 DOWN unanimity is CONSISTENT WITH a true up-rate as high as ~53%. Rule of Three gives 3/4 = 75%, safely conservative. The point: tail unanimity is a legitimate pedagogic finding but not a wide-sample effect-size claim.

Q: Where does the '3' come from?

It's just -ln(0.05) ≈ 2.996 rounded to 3. Derivation: for the true success rate p to satisfy P(observe 0 successes in n trials) = 0.05, we need (1-p)^n = 0.05, so ln(1-p) = ln(0.05)/n, so 1-p = exp(ln(0.05)/n). Taylor expansion of exp(x) at x=0 gives exp(-3/n) ≈ 1 - 3/n for small |3/n|, so 1 - p ≈ 1 - 3/n → p ≈ 3/n. The approximation is off by about 22 percentage points at n=4 (75% vs 53% exact) but drops below 1 percentage point at n=20 and below 0.2 percentage points at n=50. The 'rule of three' name — coined by Hanley & Lippman-Hand's 1983 JAMA paper 'If nothing goes wrong, is everything alright?' — is used in medical statistics for exactly this small-sample-null-count corner case.

For n Bernoulli trials with x=0 successes, the one-sided 95% upper confidence bound on the true success rate is approximately 3/n. Exact formula: p_U = 1 − 0.05^(1/n). Directly applied to today’s slot 1 (GBPJPY × Japan Prelim GDP, 0-of-4 UP at 15m): exact 95% upper CI is 52.71%. The 4-of-4 unanimity is CONSISTENT WITH a true up-rate as high as roughly 1-in-2.

Statistics for Traders #40 opens a NEW small-sample-inference thread after Stats #34-#39 closed the 6-post multivariate-classifier arc. It directly serves reader interpretation of the tail-unanimity findings that slot 1 publishes every day. Rule of Three is the classic medical-statistics tool (Hanley & Lippman-Hand, JAMA1983) for “if nothing goes wrong in n trials, is everything alright?” — and the answer is always “maybe, but the true rate could still be up to 3/n”.

Log-scale x-axis (sample size n from 3 to 100) with two curves: exact 95% one-sided upper CI curve p_U = 1 - 0.05^(1/n) (coral solid line) and the Rule of Three approximation 3/n (yellow dashed line). Both curves decay monotonically from ~0.63/1.00 at n=3 to ~0.03/0.03 at n=100. A vertical annotation line at n=4 marks today's slot 1 GBPJPY sample; horizontal dashed line at 0.527 marks the exact 95% upper CI for n=4. Annotations label 'n=4: exact 0.527, RoT 0.75' and 'n=30: exact 0.095, RoT 0.100 (converges)'. — Exact one-sided 95% upper CI (coral) vs Rule of Three approximation (yellow dashed) as a function of sample size. For n≥20 the two agree within 2 percentage points; for n≤10 the Rule of Three is conservative (safer, wider bound).

Exact vs Rule-of-Three at α = 0.05 (one-sided)

n	Exact p_U	RoT 3/n	RoT − Exact	Reading
3	0.6316	1.0000	+0.368	unanimity in only 3 trials tells you almost nothing
4	0.5271	0.7500	+0.223	TODAY'S GBPJPY SAMPLE — up to 53% true up-rate
5	0.4507	0.6000	+0.149	still very wide
6	0.3930	0.5000	+0.107
10	0.2589	0.3000	+0.041	starts feeling meaningful — below 30%
15	0.1810	0.2000	+0.019
20	0.1391	0.1500	+0.011
30	0.0951	0.1000	+0.005	unanimity now 'sharp' — below 10%
50	0.0582	0.0600	+0.002
100	0.0295	0.0300	+0.001	approximation error under 0.1 pp

Derivation — where does the 3 come from?

Under the null hypothesis that the true success probability is p_U, the probability of observing 0 successes in n trials is (1 − p_U)^n. Setting this equal to α = 0.05 (the one-sided 95% CI cutoff) and solving:

  (1 − p_U)^n = 0.05
  n · ln(1 − p_U) = ln(0.05)
  1 − p_U = exp(ln(0.05)/n)
  p_U = 1 − 0.05^(1/n)                        [EXACT]

  Taylor: exp(-3/n) ≈ 1 − 3/n  for small |3/n|
  So    p_U ≈ 1 − (1 − 3/n) = 3/n              [RULE OF THREE]

  The "3" is just −ln(0.05) ≈ 2.996, rounded.

The approximation is off by 37 percentage points at n=3 and 22 pp at n=4 (Taylor expansion breaks down at small n where 3/n isn’t small), converges to under 5 percentage points by n=10, and drops below 1 pp by n=20. For everyday use in Vantage tail-unanimity posts (typical tail-bucket sizes n=3-8 on quarterly macro events), the exact formula is worth computing; for n≥15 the 3/n rule is close enough.

Applied to today's slot 1

The GBPJPY × Japan Prelim GDP big-miss bucket at 15m has n=4, x=0 UP. The rest of the numbers follow directly:

Bound	Value at n=4	Meaning
One-sided exact 95%	52.71%	true up-rate consistent with anything from 0% to 52.71%
Rule of Three (3/n)	75.00%	conservative approximation — safer upper bound
Two-sided Clopper-Pearson	60.24%	wider than one-sided because α/2 replaces α
Wilson score (two-sided)	≈49.0%	narrower — asymptotic, doesn't guarantee exact frequentist coverage
Agresti-Coull (two-sided)	≈54.6%	pseudo-count adjusted Wald; intermediate width
Naive Wald (WRONG)	0.00%	collapses to [0,0] — do NOT use for x=0

Reading: today’s 4-of-4big-miss DOWN unanimity on GBPJPY is a legitimate pedagogic finding (the sign flip at the big_miss → small_miss threshold replicates on the fifth JPY-cross of the arc, and the 1h magnitude sets a new family record at -13.45p), but it is NOT evidence that the true up-rate is exactly zero. It’s consistent with a true rate up to 53%at 95% confidence. Trading it as a 100% reliable directional signal would overstate the sample.

Why this matters for every tail-unanimity slot-1 post

The Vantage News Impact Explorer routinely finds tail-bucket unanimities on quarterly macro events (JPY GDP, NZ CPI, AUD Trimmed Mean CPI, NZ Employment, etc.). Because the tail buckets hold at most 3-8 releases out of 15 years of data, every one of these unanimities is a small-sample finding. Rule of Three quantifies the uncertainty: an n=3 unanimity leaves the true rate anywhere up to 63.2%; an n=4 leaves it up to 52.7%; an n=6 leaves it up to 39.3%. Only at n≥15 does the upper bound drop below 20%, and n=15 would require ~4 years of monthly-release data or ~15 years of quarterly data on a rare tail bucket.

Practical implication for readers: treat every slot-1 tail-unanimity finding as a directional-consistency observation, not a probability point-estimate. Position size for the possibility that the true rate is halfway between 0% and the Rule-of-Three upper bound — that’s the correct Bayesian posterior under a uniform prior.

Series placement and verification

Stats #34-#39 formed a 6-post multivariate-classifier arc on the UK Core CPI × GBPCHF panel (MANOVA → canonical direction → LDA → Mahalanobis → QDA → multinomial logit). Stats #39 closed that arc. Stats #40 opens a NEW small-sample-inference thread with the Rule of Three, directly connected to today’s slot 1. All numbers computed by elementary algebra; exact formula p_U = 1 − α^(1/n) derived from the binomial null P(X = 0 | n, p_U) = α. Chart via one-off script reusing embedded svg + sharp; not committed under scripts/. Queue next: Fisher’s exact test (the 2x2 sibling of Rule of Three); Clopper-Pearson exact interval derivation; Rule of Five variant for x=1 (the 5/n approximation); Bayesian posterior with Beta(1,1) prior on tail-unanimity findings. Prior queue items rotate: suppressor-variable partial-vs-semi-partial (queued since Stats #25), higher-order VAR-b prewhitening (queued since Stats #20), heteroskedastic joint kurtosis (queued since Stats #33), Regularized Discriminant Analysis (Friedman 1989 RDA; queued since Stats #38).

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