Law of Small Numbers: Which Hospital Records More 60%-Boy Days?
The 15-sample process crosses 60% more often than the 45-sample process.
Precise claimIn this fixed p=.5 binomial comparison, n=15 produces a wider proportion distribution and crosses the strict >60% threshold more often than n=45.
Applies
Exact binomial calculations comparing n=15 with n=45 under p=.5, a strict proportion-above-60% event and a synthetic 365-day expectation.
Does not prove
The page does not claim the exact tail probability decreases at every adjacent integer n, model real demographic dependence, claim large samples cure bias, or treat an expected annual count as a forecast guarantee.
Portable rule
If a proportion looks extreme, then inspect its denominator and sampling process before attributing the deviation to a real underlying change.
Declare the discussion context, then copy a stable link. No account or personal data is attached.
randomness-illusion puzzle
Law of Small Numbers: Which Hospital Records More 60%-Boy Days?
Commit to the classic two-hospital judgment, inspect exact binomial tails, then keep probability and threshold fixed while changing only daily sample size.
SMALL HOSPITAL15
births every day
Same modelp = 50%Count days above 60%
LARGE HOSPITAL45
births every day
The reveal · Exact binomial tails
The smaller daily sample crosses the threshold more than twice as often.
You committed to: —
The process did not become more biased. The smaller sample simply swings farther.
n = 1515.09%
55.1 expected days / 365
n = 456.76%
24.7 expected days / 365
Strict threshold audit: 9/15 and 27/45 equal exactly 60%, so neither is counted. The tails begin at 10/15 and 28/45.
In this fixed p=.5 binomial comparison, n=15 produces a wider proportion distribution and crosses the strict >60% threshold more often than n=45.
These are model probabilities and expected counts, not a forecast that either hospital must produce that exact annual total.
The rebuilt model
You assumed:
If two hospitals share the same 50/50 process, they should record more-than-60% days about equally often.
The actual model:
The sample count X follows Binomial(n,p); the observed proportion X/n has standard deviation sqrt(p(1-p)/n), so n=45 is more concentrated around p than n=15. Exact threshold tails can jump at adjacent integer n because the cutoff count is discrete.
The variable that failed you:
Denominator — the two percentage processes share the same center but not the same sampling spread because one daily percentage is based on 15 observations and the other on 45.
Change one variable
Keep the coin and threshold. Increase only n.
p 50%threshold >60%calculator exactchanged n: 15 → 45
Reference mode opens the exact distribution without manufacturing a prediction.
Daily sample n15
P(proportion > 60%)15.09%
Proportion SD12.91 pp
>60% tail
—
This exact tail is a discrete count. It can jump at adjacent values of n when the first qualifying count changes; this page compares the stated n = 15 and n = 45 cases.
Reproduces the 45-versus-15 hospital judgment and more-than-60% threshold as a teaching example.
official-doc — checked 2026-07-21.
Scope: Exact binomial calculations comparing n=15 with n=45 under p=.5, a strict proportion-above-60% event and a synthetic 365-day expectation.
Does not prove: The page does not claim the exact tail probability decreases at every adjacent integer n, model real demographic dependence, claim large samples cure bias, or treat an expected annual count as a forecast guarantee.