# Axioms of Probability

In the last article, we discussed what is meant by Experiment, Sample Space, and Events.

Our main goal is to answer the below question

So, from a sample space, there are multiple events possible and for every subset of the sample space or for every event, we want to know the chance of that event occurring.

So, we are searching for such a function say it’s denoted by ‘P’ for probability and it takes in an event(which is a subset of the sample space), we want this function to return a number such that this number reflects the probability or the chance of this event occurring.

Now, this function must satisfy the axioms of probability.

There are 3 axioms:

1. The probability of any event should always be greater than 0
2. The probability of the sample space(as it contains all the possible outcomes) should be 1, this means that we have a cap for the probability values, it should be greater than equal to 0 and it should be less than equal to 1.
3. If there are the ‘n’ disjoint events, then the probability of the union of those ‘n’ events would simply be the sum of the probability of individual events.

If we look at the third axiom, in a way it says that we can compute the probability of larger events from smaller events.

And the smallest possible event would be just one outcome. Let’s say the sample space contains 100 outcomes, then we could think of each outcome as one event

Let’s say we roll a dice and here the possible outcomes

And given the probability of these events, we can compute the other probabilities, for example, the probability of a set B which denotes that the outcome should be an odd no. can be represented as in the below image and since the three events in this case which makes up the event B are mutually disjoint, that means we can simply add up the probability value for these 3 events

## Some properties of probability

Now that we have discussed the axioms of probability, we can look at some properties of probability.

1. Given the probability of an event A, we can derive the probability of the complement of this event

2. The probability of any event would always be less than or equal to 1, we can understand this using Property 1

3.

The proof for the same is below:

4. Say a dice is rolled, then there are 6 possible outcomes and the sum of probabilities of all outcomes is equal to 1 as at least one of the 6 numbers(possible outcomes) will show up on the dice

5. The probability of a null set(contains no element) is equal to 0

Let’s take some examples where we use the axioms, properties of probability:

Say in the game of cricket, the outcome of runs scored on any given delivery is from 0 to 6 excluding the exception cases and we call these outcomes by a particular notation: A₀ denotes the event if the run scored is 0, A₁ denotes that the run scored is 1 and so on all the way up to A₆.

Now suppose someone has given the probability for each of these outcomes

And we are interested in knowing the probability of scoring an even number of runs so that would be the set {0, 2, 4, 6} (considering 0 as well) and we can write them as the union of events A₀, A₂, A₄, A₆, and since these events are disjoint, we can apply the axiom of probability and compute the probability of the union as the sum of the probabilities of individual events

Similarly, we could solve for other scenario say the probability that the runs scored will be less than 5

We could have other scenarios as well where the two sets/events are not disjoint for example if we are interested in knowing the probability that the runs scored(value) is divisible by 2 or 3. As the keyword ‘or’ is used, we know that it will be a union of two sets, we can write the two sets individually as below:

Let’s take one more example:

If we take A₁ as the event that the ball has Type 1 defect and A₂ is the event that the ball has Type 2 defect.

Given this information, we want to know the probability that the ball bearing has Type 1 or Type 2 defect:

We could also compute the probability that the ball bearing has neither Type 1 nor Type 2 defect:

In this article, we discussed the axioms of probability and the properties of the probability function.