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Law Of Large Numbers Calculator

Work out law of large numbers instantly with clear inputs, formula shown and shareable results.

About Law Of Large Numbers Calculator

The law of large numbers says the observed relative frequency converges to the true probability as trials accumulate. The standard error √(p(1−p)/n) quantifies how fast: precision improves only as √n.

Formulas

Standard error of a proportion

SE = √(p(1 − p)/n)

Trials for a tolerance

n = z²·p(1 − p) / tolerance²

Worked example

With the default values below, Law Of Large Numbers Calculator returns P(observed rate within tolerance) of 0.79409665. Change any input above and every figure here is recalculated in your browser.

Inputs used

True probabilityaccepts 0.01–99.99 %
50 %
Number of trialsminimum 1
1,000
Tolerance around the true valueminimum 0.01 %
2 %

Results

P(observed rate within tolerance)
0.79409665
Standard error of the observed rate
0.0158113883
Trials needed for 95% confidence at that tolerance
2,401
Expected successes
500.0000

Step by step

  1. 1Values used
  2. 2Standard error of a proportion
  3. 3Trials for a tolerance
  4. 4P(observed rate within tolerance)= 0.79409665
  5. 5Standard error of the observed rate= 0.0158113883
  6. 6Trials needed for 95% confidence at that tolerance= 2,401
  7. 7Expected successes= 500.0000

Assumptions and limits

  • The result is derived only from the 3 inputs listed above. Anything not among them is not part of the calculation.
  • In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
  • Figures are rounded for display to 10 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.

How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.

Frequently Asked Questions

For a fair coin over 1000 tosses within 2%?

The standard error is 0.0158, so the observed rate lands within 2 percentage points about 79.3% of the time.

Does it mean results 'even out'?

No. The proportion converges but the absolute difference in counts typically grows — the gambler's fallacy confuses the two.

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