# Empirical Probability

Probability is the possibility that any given event will occur. Empirical probability is regarded as an objective probability, which describes the extent up to which an event has occurred in the past. Hence, it is also called the relative frequency or experimental probability. This makes empirical probability different from others. We get better results from a practical situation, as compared to a theoretical approach.

In mathematical terms, Empirical Probability is defined as the ratio of number of times a particular event occurs to the total number of trials that took place for that expected outcome to occur.

FORMULA FOR AN EMPIRICAL PROBABILITY

Empirical probability depends upon how the event had occurred during the trials. The formula for Empirical probability is given below:

P(E) = Number of times an event occursTotal number of times the experiment is performed (number of trials)

where,

Number of times an event occurs - refers to the number of times a favourable event has occurred

Total number of times the experiment is performed (number of trials) – refers to the total number of times the event was performed

However, it is to be noted that in empirical probability, the experimental conditions may differ from one experiment to the other. This is the reason that an empirical probability is regarded as an estimate of an event from a statistical point of view.

According to a famous mathematician, Thompson, empirical probability is denoted with a prime notation. This is expressed as follows:

P′(A) = n(A)n

where:

• n(A) is the total number of times the event A takes place
• n is the total number of times the experiment is done

There is a particular way to calculate empirical probability. Usually, the empirical probability is based upon the law of large numbers. If you are looking for a precise answer, you need to gather more information and then perform the entire calculation. The larger the volume of information or data, the higher is the accuracy.

ILLUSTRATION ON EMPIRICAL PROBABILITY

Let us look at this simple example that will help you understand empirical probability better.

## Eight coins are tossed in total. Frequencies of the number of tails that appeared are mentioned below.

 X 0 1 2 3 4 5 6 7 8 F 2 15 29 57 70 59 20 10 8

Determine the probability of getting tails:

i) Less than 4 times

ii) Equal to 5 times

iii) More than 6 times

Total number of heads that occurred in this experiment = 2 + 15 + 29 + 57 + 70 + 59 + 20 + 10 + 8 = 270

i)  P (occurred tails less than 4)

As per the given table, we could see that = 2 + 15 + 29 + 57 = 103

Thus, P (tails less than 4) = 103/270 = 0.38

ii) P (occurred tails equal to 5).

From given table, the required number of cases = 59

Therefore, P (tails equal to 5) = 59/270 = 0.22

iii) P (more than 6 tails)
Total number of cases = 18
So, P (tails more than 6) = 18/270 = 0.07

### Important Concepts of Maths

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