# Empirical Probability

## Trigonometry # Empirical Probability

Empirical probability is a type of experimental probability that depends on past data or historical data. Empirical probability is the likelihood of an event to occur based on some previous years data.

If I were to ask you which cricket team had a greater chance of beating India- Australia or South Africa, what would be your answer? There is no way for you to precisely determine and compare the chances of the two events happening.The concept of chance and possibility of an event intrigued the mathematicians of the age.

The entry of mathematics into the field of possibility and chance was spurred by card players and gamblers. On one such occasion, a gambler walked up to the famous mathematician, Pierre de Fermat and asked his help for improving his chances of winning. This led to the development of Probability Theory. Probability is closely connected with chance. With the help of mathematics and some clever mathematicians, we were able to describe the changes or the possibility of an event occurring with numbers (more accurately, Ratios). What is Probability? What is Empirical probability?

A lot of techniques were tried to describe and measure the possibility of an event occurring with no success. When all else fails, Mathematics is our only true hope.

## What is Empirical Probability?

Experimental or empirical probability is the probability of an event based on the results of an actual experiment conducted several times. In theoretical probability, we assume that the probability of occurrence of any event is equally likely and based on that we predict the probability of an event. For example: when we toss an unbiased coin, the chances of occurrence of head or tail is equally likely. So, the probability of occurrence of head is ½ or 50%. Empirical probability or experimental probability is based on actual experiments and adequate recordings of the occurrence of events.

Actual experiment is conducted to determine the probability of occurrence of an event. Experiments not having fixed results are known as random experiments and the outcome of such experiments are uncertain. Random experiments are repeated multiple times to determine its likelihood.  The number of times an experiment is repeated is better described as number of trials.

## Empirical Probability Formula

Mathematically, the formula for emperical probability can be given as:

$$Experimental\; Probability = \frac{Number\; of\; times\; an\; event\; occurs}{Total\; number\; of\; trials}$$

## Solved Example on Empirical Probability

Example: A fair die was rolled 120 times, find the number of time 5 turned up.

Solution: As we know each number has equal probability of occurrence, i.e. $$\frac{1}{6}$$.
The probability of occurrence of event is $$120 \times \frac{1}{6} = 20$$

Therefore, the occurrence of 5 is 20 out of 120 (on average).

The experimental or empirical probability of an event is based on what has actually happened while the theoretical probability of the event attempts to predict what will happen on the basis of the total no. of outcomes possible. As the number of trials in an experiment, go on increasing we may expect the experimental and theoretical probabilities to be nearly the same.

## Frequently Asked Questions on Empirical Probability

### What is empirical probability?

Empirical probability is the experimental probability that depends on past data or historical data.

### How to find the empirical probability?

The empirical probability formula is:
P(E) = f/n
where, f is number of times events occur and n is the total number of times the experiment performed.

### What is the example of empirical probability?

Roll a die three times and see the corresponding result. The empirical probability of rolling a 4 is 0% (0/3).

### What is the difference between theoretical probability and empirical probability?

Theoretical probability is the probability based on assumption that the events are going to happen whereas empirical probability is based on the observations of the occurred events.