  Course Content

Probability Theory Mastering

##   Law of Large Numbers for Bernoulli Process

A Bernoulli trial is a statistical experiment with only two possible outcomes, usually success and failure, with fixed probabilities of occurrence on each trial. It was considered in more detail in the Probability Theory Basics course.

In a Bernoulli process, each trial is independent, meaning the outcome of one trial does not affect the outcome of any other trial. The probability of success, denoted by p, is the same for every trial. The probability of failure is indicated by q = 1 - p. Let's try to apply the law of large numbers to this scheme. Assume that we provide n experiments and want to calculate the total number of successful results. According to the law of large numbers law, we can do it as follows: Each variable in the numerator represents the result of one experiment: if the experiment is successful, this variable equals 1 ( with the corresponding probability of success p), and if the experiment is not successful, this variable equals 0 with probability 1-p. It is obvious that the conditions of the law of large numbers are satisfied in this case: all these variables are independent (since the experiments are carried out independently), equally distributed, and have a finite distribution (this can be seen from the distribution series).
Thus, using the law of large numbers, we can estimate the probabilities of an event's occurrence by analyzing the occurrence frequency of an event.
Let's look at an example: suppose we flip a deformed coin with a displaced center of gravity (head and tail probabilities are different). Our task is to estimate the probability of the head falling out. Look at the code below:  Similarly, the law of large numbers can be generalized for a polynomial scheme: for 1, we consider the occurrence of the event/events of interest to us, and for 0, all other results. Let's look at an example:  