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 for a 30% chance of rain or considering the odds of winning a game. It helps us navigate a world full of uncertainty by putting a number on it.

This number always falls somewhere between 0 and 1.

• A probability of 0 means the event is impossible. The chance of a standard six-sided die landing on a 7 is 0.
• A probability of 1 means the event is certain. The chance that the sun will rise tomorrow is, for all practical purposes, 1.
• A probability of 0.5 means the event is just as likely to happen as not to happen, like getting heads on a fair coin flip.

To calculate a probability, we first need to know all the possible things that could happen. This complete set of all potential outcomes is called the sample space.

For a single six-sided die, the sample space is {1, 2, 3, 4, 5, 6}. For a coin flip, it's {Heads, Tails}. It's a simple list of everything that could occur.

Within the sample space, we're usually interested in a specific outcome or a group of outcomes. This is called an event.

An event can be simple or complex. For our die-rolling experiment, here are a few possible events:

• The event of rolling a 3 is the set {3}.
• The event of rolling an even number is the set {2, 4, 6}.
• The event of rolling a number greater than 4 is the set {5, 6}.

Each of these is a subset of the full sample space {1, 2, 3, 4, 5, 6}.

Probability isn't arbitrary; it follows three simple but powerful rules called axioms. These are the foundational truths that make all of probability theory work. Think of them as the rules of the road for calculating and interpreting probabilities. Let's look at them one by one.

The first axiom states that the probability of any event must be a number greater than or equal to zero. You can't have a negative chance of something happening. It either happens, or it doesn't.

The second axiom tells us that the probability of the entire sample space is 1. This makes sense: if you roll a die, it is a certainty (a probability of 1) that one of the possible outcomes—1, 2, 3, 4, 5, or 6—will occur. One of the possibilities must happen.

What if we want to know the probability of one of several different things happening? For example, what's the probability of a die roll being a 1 or a 2?

The third axiom helps us. It says that if you have two or more events that cannot happen at the same time (called mutually exclusive events), the probability of any of them happening is the sum of their individual probabilities.

Rolling a 1 and rolling a 2 are mutually exclusive because a single roll can't be both at once. 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 axioms are the building blocks for all of probability. They ensure our calculations are logical and consistent, giving us a reliable framework for reasoning about uncertainty.

Now that we've covered the basics, let's test your understanding.