Law of Large Numbers for Bernoulli Process | The Limit Theorems of Probability Theory
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 outcome of one experiment: it's `1` if the experiment succeeds (with probability `p`) and `0` if it fails (with probability `1-p`).

In this case, the conditions of the law of large numbers are met: the variables are independent (as the experiments are independent), identically distributed, and have a finite expectation (as shown by the distribution series).

Therefore, we can use the law of large numbers to estimate the probabilities of an event's occurrence by analyzing the frequency of its occurrence.

For example, let's consider flipping a coin with a displaced center of gravity. Our goal is to estimate the probability of it landing heads up. Check out 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:

Can we say that the frequency of occurrence of a certain event can be interpreted as the probability of this event if the frequency is calculated from a fairly large number of experiments?

Everything was clear?

Section 2. Chapter 2

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 outcome of one experiment: it's `1` if the experiment succeeds (with probability `p`) and `0` if it fails (with probability `1-p`).

In this case, the conditions of the law of large numbers are met: the variables are independent (as the experiments are independent), identically distributed, and have a finite expectation (as shown by the distribution series).

Therefore, we can use the law of large numbers to estimate the probabilities of an event's occurrence by analyzing the frequency of its occurrence.

For example, let's consider flipping a coin with a displaced center of gravity. Our goal is to estimate the probability of it landing heads up. Check out 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:

Can we say that the frequency of occurrence of a certain event can be interpreted as the probability of this event if the frequency is calculated from a fairly large number of experiments?