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A rare-event sample mean at thirty observations

Thirty independent Bernoulli observations with event probability 0.001 produce a zero-event sample with about 97.0431% probability.

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Specify an independent finite-variance model

Each synthetic observation is one with probability 0.001 and zero otherwise. The observations are independent and share that distribution. Their mean is 0.001 and variance is 0.001 × 0.999. This model meets the classical finite-variance conditions.

Calculate the mass at zero

Let K count events in thirty observations. The sample mean is K/30, and P(K = 0) = 0.999^30 ≈ 0.9704309673. Thus nearly all samples have mean zero, with remaining outcomes at positive multiples of 1/30. The distribution is discrete and strongly asymmetric at this n.

Exact first outcomes
KSample meanProbability
000.97043097
10.033333333333333330.029142071
20.066666666666666670.00042298301
30.10.0000039517933

Inspect the complete count distribution

All count values zero through thirty are included. Most probability lies at zero; the small positive outcomes are not evenly balanced by negative outcomes because the count cannot be negative. A normal curve at n = 30 would not reproduce this mass pattern.

00.48520.970401530Event countProbability00.48520.970401530Event countProbability
Rare-event count in thirty independent trials

Exact binomial probabilities for a fictional event probability of 0.001; all counts from zero to thirty are included. This is an illustrative example.

View the chart values
Rare-event count in thirty independent trials: underlying values
SeriesEvent countProbability
Exact probability00.97043097
Exact probability10.029142071
Exact probability20.00042298301
Exact probability30.0000039517933
Exact probability42.6701306E-8
Exact probability51.3898578E-10
Exact probability65.7968709E-13
Exact probability71.9894881E-15
Exact probability85.7255037E-18
Exact probability91.4009685E-20
Exact probability102.9449789E-23
Exact probability115.359867E-26
Exact probability128.494951E-29
Exact probability131.1774014E-31
Exact probability141.4311328E-34
Exact probability151.5280697E-37
Exact probability161.4339994E-40
Exact probability171.1821228E-43
Exact probability188.5460996E-47
Exact probability195.4029395E-50
Exact probability202.9745913E-53
Exact probability211.41789E-56
Exact probability225.8062652E-60
Exact probability232.0215921E-63
Exact probability245.9022124E-67
Exact probability251.4179489E-70
Exact probability262.7295544E-74
Exact probability274.0478322E-78
Exact probability284.3413044E-82
Exact probability292.997E-86
Exact probability301E-90

Separate this finite sample from the limit

The standardized mean approaches normality as n grows for this fixed model. That limit does not certify approximation quality at thirty. A report should assess the actual model and approximation needed for its task rather than declaring any thirty observations normal.

Further references