Empirical and Theoretical Probabilities
In order to distinguish between theoretical and empirical probabilities, we have to give definitions of both. Empirical probability is a probability is the estimation of the observing result of a real experiment. It is based on the number of favorable outcomes of a particular event and the total number of trials. The simplest example that demonstrates the concept of empirical probability is given below:
Consider the experiment, when you roll two dice 10 times in a row. You record the sum of values on each die. The results are: 7, 10, 5, 4, 11, 7, 6, 9, 7 and 12. What is the probability of rolling a 3? There was no such value in any of 10 trials. Hence, empirical probability is:
P=number of event occurstotal number of observed occurances=010=0
Theoretical probability is the ratio between number of ways when even occurs and the total number of possible outcomes. Here we say about the sample space with equally likely and known outcomes. Referring to the example, given above, the theoretical probability of rolling a sum 3 on the two dice is 2/36, because there are two favorable events: when 1 is on the first die and 2 is on the second, and verse versa.
In restaurant business, the difference between theoretical and empirical probabilities can be demonstrated in the following example. Assume you have 100 different items in the menu. You know that a restaurant inspector comes to your restaurant once per month and randomly orders one item for evaluation. You consider a period of 12 months. An inspector came to you 12 times and ordered items with the following numbers: 2, 17, 33, 61, 17, 14, 69, 56, 93, 44, 17 and 23. What is empirical probability that inspector orders item #17? This item was ordered 3 times during a 12 months period. Hence, empirical probability is 3/12 = ¼. However, theoretical probability of this event is 1/100 (because one item is being chosen at a random from 100 possible items).
The law of large number is related to this discussion in the following way: as the number of trials increases, the value of empirical probability of an event approaches to the value of theoretical probability.
