Introduction
We frequently use the term Probability but don't realize how powerful this concept is. In simple terms, probability is the probability that something will happen. And one of the fundamental concepts of probability is the axioms of probability, which are essential for statistics and exploratory data analysis.
Axioms mean a rule, a principle that most people believe to be true. It is the premise on the basis of which we make further reasoning
In this article, I'll cover the three axioms of probability in detail.
Note: If you are more interested in learning concepts in an audiovisual format, we have this full article explained in the video below. If that is not the case, you can keep reading.
Probability axioms
There are three axioms of probability that form the basis of probability theory:
Axiom 1: Event probability
The first is that the probability of an event is always between 0 Y 1. 1 indicates a defined action of any of the results of an event and 0 indicates that an event result is not possible.
Axiom 2: Probability of the sample space
For the sample space, the probability of the entire sample space is 1.
Axiom 3: Mutually exclusive events
And the third is: the probability that the event contains any possible outcome of two mutually disjoint is the sum of their individual probability.
Now let's see each of them in detail!!
1. Probability of the event
The first axiom of probability is that the probability of any event is between 0 Y 1.
As we know, the probability formula is that we divide the total number of outcomes in the event by the total number of outcomes in the sample space.
And the event is a subset of the sample space, so the event cannot have more results than the sample space. Clearly, this value will be between 0 Y 1 since the denominator is always greater than the numerator.
2. Probability of the sample space
The second axiom is that the probability of the entire sample space equals 1.
Let's take an example from the data set. Suppose we have to find out the probability that clients move by their type of occupation.
In our data set, have 4 clients, one of them salaried and three of them autonomous. The salaried woman is going to beat. As we only have one salaried woman who is going to leave, the number of salaried clients who will not leave is 0. Between the 3 autonomous clients, two are going to abandon and we can see that one is autonomous the female is not going to beat. This is the complete data set:
Then, the probability of a client's agitated state by profession, in the sample space of the problem, we actually have:
salaried churn, salary no churn, churn on your own, churn on your own, no churn
And as we discussed its distribution previously, in this customer sample space:
Salaried turnover = 1
Employee No abandonment = 0
Churn of freelancers = 2
Self-employed No dropout = 1
If I were to find out the probability that a person who is a woman receives a salary and is beating, would be equal to:
Similarly, the probability that wages are not lost is:
Then we have the Churn of freelancers:
Y, Finally, freelancers who do not leave:
And if we add them all we get 1:
Basically, say that this is our complete sample space and that the total probability that we get here is equal to 1. This brings us to the axiom 3, which is related to mutually exclusive events.
3. Mutually exclusive event
If you remember the join formula, you will remember that the intersection term is not here, which means that there is nothing in common between A and B. Let's understand this particular type of event which is called Mutually exclusive events.
These mutually exclusive events mean that such events cannot occur together or, in other words, they have no common values or we can say that their intersection is zero / null. We can also represent such events as follows:
This means that the intersection is zero or they have no common value. For instance, if he
Event A: gets a number greater than 4 after rolling a dice, the possible results would be 5 Y 6.
Even B: gets a number less than 3 when throwing a dice. Here the possible results would be 1 Y 2.
Clearly, these two events cannot have a common outcome. An interesting thing to note here is that events A and B do not complement each other., but they are mutually exclusive.
Mutually exhaustive
A more important concept is Mutually exhaustive event which is often confused with mutually inclusive events. Mutually exhaustive events mean that such events together constitute all that can possibly happen in a random experiment.. That means that the union of these events makes the sample space:
Let's understand this with an example:
Event A: Get a number greater than 2 after a die, the possible results would be:
Event B: Get a number less than 4 after rolling a dice. Here the possible results would be:
Clearly, these two events together constitute all the outcomes that can take place after rolling a die.
How do mutually exhaustive events differ from mutually exclusive events??
In the example above, getting a number 3 was common between both events. Therefore, these CANNOT be mutually exclusive, but they are definitely mutually exhaustive. Secondly, if we have another event:
Event C: Get a number less than 3 after rolling a dice, the possible results would be:
Now we can say that Event A and Event C are mutually exclusive since they have nothing in common.
Final notes
In this article, we cover the axioms of probability and the difference between mutually exclusive and mutually exhaustive events.
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