UPSKILL MATH PLUS

Learn Mathematics through our AI based learning portal with the support of our Academic Experts!

5. Introduction to empirical approach

Empirical approach

When a probability is found considering the previously observed data or based on a larger set of observations, it is called an empirical probability approach.

Example:

When a \(20000\) pairs of shoes are ordered from the company and \(2500\) of them are faulty, what is the number of faulty pair of shoes?

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:

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

Example:

Continuing with the situation discussed earlier:

The number of trials, \(n = 20000\)

The number of faulty pair of shoes, \(r = 2500\)

Then, \(P(E) = \frac{2500}{20000}\)