1.5.2 Axiom of Normalization
This axiom states that the probability of the sample space is equal to 1.
Let's recall that represents the set of all possible outcomes in the experiment, and an event represents a subset of that sample space. In the normalization axiom, our event is the sample space itself, so we want to find the probability of "obtaining an outcome from all the possible outcomes of the sample space." As a result, the event encompasses the entire set of the sample space, rather than being limited to a subset of the sample space. It is important to note that an event can still be a proper subset of the sample space in other cases, but in the context of normalization, we consider the event to be the entire sample space.
In the sample space, we have all the outcomes, and one of these outcomes will certainly occur. When something is absolutely certain, it has a 100% chance or a probability of 1. Therefore, the probability assigned to the event, which now represents the entire sample space, is equal to 1.
What we have established here is:
Example 1.25. Let's revisit the example of rolling a die and examine the key elements involved in the context of the normalization axiom. The sample space is equal to the set of all possible outcomes of a die i.e. {1, 2, 3, 4, 5, 6}, Event is “rolling any one number from the entire set of the sample space”. So, what do you think is the probability that the die will land on any of the six numbers?
The die has six sides and we can say with absolute certainty that the die will land on any one of the numbers. We capture this certainty by saying that the probability that the event is going to occur is equal to 1.
