Theorem

Let AA denote an event with a probability pp to occur in a single trial. Then, if kk is the number of occurrences of AA in nn independent trials,

P(|kn−p|>ε)<pqnε2P\left(\left\lvert \frac{k}{n} - p \right\rvert > \varepsilon\right) < \frac{pq}{n\varepsilon^2}

Alternatively, this is described in a limit form:

limn→∞⁡(|kn−p|≥ε)=0\lim_{n \to \infty} \left(\left\lvert \frac{k}{n} - p \right\rvert \geq \varepsilon\right) = 0

or

limn→∞⁡(|kn−p|<ε)=1\lim_{n \to \infty} \left(\left\lvert \frac{k}{n} - p \right\rvert < \varepsilon\right) = 1

or simply

limn→∞⁡kn=p\lim_{n \to \infty} \frac{k}{n} = p

See also

References

  1. A. Papoulis, U. Pillai. Probability, random variables, and stochastic processes, McGraw-Hill, 2002, pp. 58-60.
  2. https://proofwiki.org/wiki/Bernoulli's_Theorem
  3. https://link.springer.com/referenceworkentry/10.1007/978-0-387-32833-1_28