The weak law of large numbers is a result in probability theory also known as Bernoulli’s theorem. Let P be a sequence of independent and identically distributed random variables, each having a mean and standard deviation.

## Formula

$${ 0 = \lim_{n\to \infty} P \{\lvert X – \mu \rvert \gt \frac{1}{n} \} \\[7pt]

\ = P \{ \lim_{n\to \infty} \{ \lvert X – \mu \rvert \gt \frac{1}{n} \} \} \\[7pt]

\ = P \{ X \ne \mu \} }$$

Where −

- ${n}$ = Number of samples
- ${X}$ = Sample value
- ${\mu}$ = Sample mean

### Example

**Problem Statement:**

A six sided die is rolled large number of times. Figure the sample mean of their values.

**Solution:**

Sample Mean Calculation

\ = \frac{21}{6}, \\[7pt]

\, = 3.5 }$

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