Third Axiom

We hear the term probability a lot, but we don’t appreciate how powerful it is. The probability, in simple terms, is the likelihood or chance of something happening. The Axioms of Probability, which are necessary for statistics and exploratory data analysis, are one of the core principles of probability.

Axioms are rules or principles that the majority of people feel are true. It is the foundation around which we build our arguments.

Axioms of probability:

The base of probability theory is built on three axioms of probability:

Axiom 1: Event Probability

The first is that an event’s probability is always between 0 and 1. 1 denotes definite action of any of the event’s outcomes, while 0 indicates that no event outcomes are feasible.

Axiom 2: Probability of sample spaces

The probability of the entire sample space is 1 in sample space.

Axiom 3: Mutually exclusive event

The probability of an event comprising every potential outcome of two mutually discontinuous events is equal to the sum of their individual probabilities.

Probability of event:

The first postulate of probability is that any occurrence has a probability between 0 and 1.

For any event E, 0 ≤ P(E) ≤ 1

As we all know, the probability formula is to divide the total number of outcomes in an event by the total number of outcomes in the sample space.

p(event) = count of outcomes in event/count of outcomes in sample space.

Because the event is a subset of the sample space, it can’t have any more outcomes than the sample space. Because the denominator is always bigger than the numerator, its value must be between 0 and 1.

Mutually Exclusive Event:

p( A ∪ B ) = p(A) + p(B) for mutually exclusive event.

If you recall the union formula, you’ll see that the intersection term is missing, implying that A and B have nothing in common. Let’s look at this form of event, which is referred to as Mutually Exclusive Events.

Mutually exclusive events are those that cannot happen at the same time, or in other words, they have no shared values, or their intersection is zero/null. Such occurrences can alternatively be represented as follows:

 P(A∩B) = 0

This indicates that the intersection is 0 or that no common value exists. For instance, if the

Event A:

After rolling a die and getting a number greater than 4, the possible results are 5 and 6.

Event B:

means rolling a die and getting a number fewer than three. The probable outcomes, in this case, are 1 and 2.

Clearly, neither of these occurrences can have the same consequence. It’s worth noting that events A and B are not complementary to one another, but they are mutually exclusive.

Mutually Exhaustive:

Another key idea is Mutually Exhaustive Event, which is frequently mistaken for mutually inclusive events. Mutually exhaustive events are those that make up the entirety of what could happen in a random experiment. That is, the union of these events forms the sample space:

P(U Eni=1) = S

As an illustration, consider the following:

Event A: After rolling a die and getting a number larger than 2, the following are probable outcomes:

Event B:  Rolling dice and getting a number less than four. Here are some probable outcomes:

Clearly, both of these events add up to the total number of possible outcomes after rolling a die.

How mutually exhaustive events differ from mutually exclusive events:

Getting a number 3 was common in both instances in the previous example. So, while these are not mutually exclusive, they are mutually exhaustive. If, on the other hand, another event occurs:

Event C:  If you roll a dice and get a number less than 3, the following are probable outcomes i.e 1 and 2.

Because they have nothing in common, we can now claim that Events A and C are mutually exclusive.

Conclusion:

If two events A and B are mutually exclusive, then the probability of either A or B occurring is the probability of A occurring plus the probability of B occurring. The probability, in simple terms, is the likelihood or chance of something happening. Axioms are rules or principles that the majority of people feel are true. It is the foundation around which we build our arguments. The first is that an event’s probability is always between 0 and 1.

The probability of an event comprising every potential outcome of two mutually discontinuous events is equal to the sum of their individual probabilities.

Mutually exclusive events are those that cannot happen at the same time, or in other words, they have no shared values, or their intersection is zero/null. 

Another key idea is Mutually Exhaustive Event, which is frequently mistaken for mutually inclusive events. Mutually exhaustive events are those that make up the entirety of what could happen in a random experiment.