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Empirical Probability

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Probability is the possibility of occurrence of any event. Probability can be classified as Theoretical Probability or Empirical Probability. Now, we will look at the definition of Empirical probability. Empirical Probability can be defined as the prediction of occurrence of an event, which is calculated on the basis of collection of empirical evidence. An empirical probability is closely related to the relative frequency of any given event in a given probability distribution.

We also call empirical probability as experimental probability.

To define empirical probability, we can also say that it is the ratio of the number of "favorable" outcomes to the total number of trials. These trials of the experiments are not in a sample space. But, in an actual sequence of experiments conducted in the space, we can also say that empirical probability helps to estimate the probability using collection of observations and the experiments conducted. So, we can also say that empirical probability is the estimated probability, which takes into account the occurrence of the events.

To study empirical probability definition, P(E) of any event ‘E’ is the fraction of times, we expect any event ‘E’ to occur. If we look at the formula for calculating the empirical probability formula, we can find it as follows:

If ‘E’ is any event and P(E) is the probability of the outcome ‘s’, then

P(E) = $\frac{\text{Number of times of occurrence of event ‘E’}}{\text{Total number of observed occurrences}}$

Here, the numerator indicates the number of ways the specific event occurs and the denominator indicates the number of ways the experiment could be conducted.

Empirical probabilities are based on tests of the experiments which we conduct. Suppose, if we want to determine the empirical probability of rolling a number six on a die in three rolls, we will actually first roll the dice three times in real sense and then, count probability on my findings done after the experiment. If this Empirical Probability is compared with the Theoretical probability, it would be to simply look at the dice and see one out of six times itself in all the three times. The difference is the theoretical probability is based on theory and not actual facts collected by the events. Empirical probability is based on real testing events. If we observe the difference between the theoretical probability and the empirical probability, then we observe that the theoretical probabilities are our calculation of what "should" happen, when you are able to describe all the equally likely outcomes and on another hand, we find that empirical probability is based on the collection of real facts and figures.

If a survey in a class of 60 students is conducted to select the breed of dog of their choice and the following outcomes are figured:

Dog Number of students
BOXER 35
SPANIEL 15
OTHERS 10

Then, the Empirical Probability of choosing the BOXER Dog will be $\frac{35}{60}$, where 35 is the number of students, who had shown the choice of BOXER dog and 60 is the total strength of the class.

Empirical Probability Formula

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Empirical probability formula has a wide application in probability. With the help of this formula, we can find the probability of a particular event, when events are given in tabular form. This formula is mainly used to find the probability of empirical event which are biased on direct observation or on random experiment. There is a particular formula for finding the Empirical Probability which is given below,

P(E) = $\frac{\text{Number of times 'E' occurs}}{\text{Total number of times observed}}$

P(E) = Probability of the event 'E' will occur.

Numerator of the given probability tells the number of times and numbers of ways the particular event had occurred. Denominator of the given probability tells the sum of particular events that occur in the event.

So, by this formula, we can say that if we divide the particular event with total event, then we will find the empirical probability of the event. Generally, we find this probability for tabular type of problem with the help of this formula and we can easily find the probability of that event.
 
If we take a survey of student of different countries and we have to find the probability of the students who like mathematics and we found, in India 40 students are interested, in USA 42 students are interested, in Japan 37 student are interested and in Spain 32 student are interested.

Now, we need to find the empirical probability of the student interested in mathematics from USA. For finding this probability, we need to sum up all the number of student from all the countries and we will get 40 + 42 + 37 + 32 = 151.

Now, we will put all the things in the formulas to get the value of empirical probability.

As we know that the number of student in USA is 42, the required probability is $\frac{42}{151}$ = 0.278.

In this way, we can find the empirical probability of an event.