5. Introduction to empirical approach

The empirical probability is considered in the case of a large number of trials.

 

How to calculate the empirical probability:

 

Let \(P(E)\) be the probability of an event. For a large number of trials, \(n\), the number of probable outcomes is \(r\).

 

In this case, \(P(E) = \frac{r}{n}\).

 

We know that the probability will always be between 0 and 1.

 

This can also be written as \(0 \leq P(E) \leq 1\).

 

Let us look at the construction of the above given expression in detail.

 

Firstly, we know that \(r\) cannot be greater than \(n\). In case \(r\) is greater than \(n\), the probability value becomes greater than \(1\). It defies the primary rule of probability.

 

Therefore, \(P(E) =\) \(\frac{r}{n}\) \(< 1 \longrightarrow (1)\)

 

If \(r = 0\):

 

\(P(E) =\) \(\frac{0}{n}\) \(= 0\) \(\longrightarrow (2)\)

 

If \(r = n\):

 

\(P(E) = \frac{n}{n} = 1 \longrightarrow (3)\)

 

From \((1)\), \((2)\), and \((3)\), it is clear that:

 

\(0 \leq P(E) \leq 1\)