The probability of an event represents the frequency, or chance, that the event will happen.

For notation, if A is an event, P(A) is the probability of the occurrence of the event.

We can define the actual probability of an event, A, as follows:

Here, A is the event in question. Think of an entire universe of events where anything is possible, and let's represent it as a circle. We can think of a single event, A, as being a smaller circle within that larger universe, as shown in the following diagram:

Let's now pretend that our universe involves a research study on humans and the A event is people in that study who have cancer.

If our study has 100 people and A has 25 people, the probability of A or P(A) is 25/100.

The maximum probability of any event is 1. This can be understood as the red circle grows so large that it is the size of the universe (the larger circle).

The most basic examples (I promise they will get more interesting) are coin flips. Let's say we have two coins and we want the probability that we will get two heads. We can very easily count the number of ways two coins could end up being two heads. There's only one! Both coins have to be heads. But how many options are there? It could either be two heads, two tails, or a heads/tails combination.

First, let's define A. It is the event in which two heads occur. The number of ways that A can occur is 1.

The sample space of the experiment is {HH, HT, TH, TT}, where each two-letter word indicates the outcome of the first and second coin simultaneously. The size of the sample space is four. So, P(getting two heads) = 1/4.

Let's refer to a quick visual table to prove it. The following table denotes the options for coin 1 as the columns, and the options for coin 2 as the rows. In each cell, there is either True or False. A True value indicates that it satisfies the condition (both heads), and False indicates otherwise:

So, we have one out of a total of four possible outcomes.