# Empirical Probability – Explained with Examples

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Empirical Probability is defined as the probability of occurrence of an event calculated on the basis of past experience or experimental data. It is also known as experimental probability.

Example 1: The most well known application of the concept of empirical probability is in the area of life insurance. Suppose that we wish to estimate the probability that a man living in a particular city dies between the ages 70 to 75. The only way to calculate such a probability is on the basis of past data. Suppose that we have the data of the births and deaths which have occurred in that city. So if there have been 100 deaths in the city in the past year out of which 30 of the deaths were of people between the ages 70 and 75, we can conclude that there is a 30% chance that the man will die between the ages 70 and 75.

Example 2: Suppose that we want to calculate the probability that a bulb produced by a particular company lasts for at least six months. The experimental approach is the only way to calculate the required probability. To calculate the desired probability we should select a random sample of say, a thousand bulbs and check how many of them have a lifetime of at least six months. If 950 bulbs out of 100 last for six months then we can conclude that the bulbs produced by that company have a 95% chance of lasting for at least six months.

### Empirical Probablity Formula:

We use the relative frequency approach to define empirical probability. Suppose that a coin is tossed 100 times out of which heads occurs 40 times. We might conclude that the probability of getting heads on tossing a coin is 0.4. But it is reasonable to expect that if we repeatedly toss the coin a large number of times, we should get an approximately equal number of heads and tails. As the coin is tossed more and more times, the empirical probability will “approach” the true value of 0.5.

If an event occurs ‘a’ times out of ‘n’, then the empirical probability can be calculated using the formula,

\text{Empirical Probability = }\lim_{n\to \infty} \frac{a}{n}

The fact that the number of trials ‘n’ goes to infinity in the definition tells us a very important fact. It tells us that the concept of empirical probability is a long-term concept. We need to repeat the trial a large number of times before we can use the data to calculate the experimental probability. In practice, since it is impossible to repeat an experiment an infinite number of times we repeat the trial, a large but finite number of times.

Empirical probability is also known as a posteriori probability because it can be calculated only after (“posterior”) the experiment has been done and the data has been collected.

### Empirical Probability vs Classical/Theoretical Probability:

The classical/theoretical approach to probability consists of dividing the possible outcomes into cases that are equally likely. For example, when a dice is thrown there are six equally likely outcomes. But kind of simple enumeration is not possible in many cases. For example, we cannot calculate the probability that a man dies at a certain age using the classical approach whereas, it can easily be calculated using the empirical approach on the basis of past data.

The classical/theoretical probability is also known as a priori probability and is normally encountered in problems dealing with games of chance, such as dice rolls and playing cards. The classical probability is deductive in nature whereas, the empirical probability is inductive in nature.

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