Probability is simply a way to measure how likely something is to happen. We use it every day, whether we're checking the weather forecast or deciding if we should buy a lottery ticket. It helps us navigate a world full of uncertainty by putting a number on it.

This number always falls on a scale from 0 to 1. An event with a probability of 0 is impossible, like a standard coin landing perfectly on its edge. An event with a probability of 1 is certain, like the sun rising tomorrow morning.

Most events fall somewhere in between. A coin flip has a 0.5 probability of landing on heads, meaning it's just as likely to happen as not.

We often express the probability of an event, let's call it $A$, using the notation $P(A)$. So, for a fair coin flip, we would write $P(\text{Heads}) = 0.5$.

To calculate a probability, we first need to understand all the possible outcomes. In probability, any process with an uncertain result, like rolling a die or drawing a card, is called an experiment.

The sample space is a complete list of everything that could happen. If you roll a standard six-sided die, the sample space, often denoted by $S$, is:

Now, let's say you're interested in a specific result, like rolling an even number. This is called an event.

An event is just a collection of one or more outcomes from the sample space. The event of rolling an even number corresponds to the set {2, 4, 6}. For a simple experiment where each outcome is equally likely, the probability of an event is the ratio of the number of favorable outcomes to the total number of outcomes in the sample space.

For our die roll, the probability of getting an even number is:

Probability isn't just a set of convenient rules; it’s a formal mathematical system built on three fundamental truths called axioms. These were established by the Russian mathematician Andrey Kolmogorov in the 1930s and form the bedrock of modern probability theory.

These axioms ensure that our calculations are logical and consistent. Let's break them down.

When you roll a die, you are guaranteed to get one of the numbers from 1 to 6. The probability of rolling a number that is in the set {1, 2, 3, 4, 5, 6} is certain, so its probability is 1.

For example, when rolling a die, the event of rolling a 1 and the event of rolling a 2 are mutually exclusive; you can't do both on a single roll. The event $A$ is 'rolling a 1' and event $B$ is 'rolling a 2'.

The probability of rolling a 1 is $1/6$, and the probability of rolling a 2 is $1/6$. Therefore, the probability of rolling a 1 or a 2 is:

These three simple axioms are all we need to build the entire framework of probability, allowing us to tackle much more complex problems.

These foundational concepts—the 0-to-1 scale, sample spaces, events, and the three axioms—are the building blocks for exploring the fascinating world of chance and likelihood.