Probability is a way to measure uncertainty. It tells us the likelihood of an event happening, assigning it a number between 0 and 1. An event with a probability of 0 is impossible, while an event with a probability of 1 is certain to happen.

For many simple situations, we can calculate probability by counting outcomes. The basic idea is to compare the number of ways a specific event can happen to the total number of possible outcomes.

Think about rolling a standard six-sided die. There are six possible outcomes: 1, 2, 3, 4, 5, or 6. What's the probability of rolling a 4? There is only one way for that to happen, out of six total possibilities.

In probability, we often work with quantities whose values aren't fixed. Instead, they depend on the outcome of a random process. These are called random variables.

A random variable isn't like the variables you see in algebra, where x has a single, specific value to solve for. It's a variable that can take on different numerical values based on chance.

Let's use an example. Imagine you flip a coin twice. The number of heads you get is a random process. We can define a random variable, let's call it $H$, to represent the number of heads.

The possible outcomes of flipping a coin twice are:

• Tails, Tails (TT)
• Tails, Heads (TH)
• Heads, Tails (HT)
• Heads, Heads (HH)

The value of our random variable $H$ depends on which of these outcomes occurs. It could be 0 (for TT), 1 (for TH or HT), or 2 (for HH).

To make sure our calculations are consistent and logical, probability follows three fundamental rules, known as axioms. These were established by the mathematician Andrey Kolmogorov and form the foundation of modern probability theory.

Axiom 1: Non-negativity The probability of any event is always a non-negative number. It can be zero, but it can't be less than zero. This makes sense—you can't have a negative chance of something happening.

Axiom 2: Unit Measure The probability of the entire sample space—that is, the set of all possible outcomes—is 1. If we consider every single possibility, it's certain that one of them will occur.

For our six-sided die, the sample space $S$ is {1, 2, 3, 4, 5, 6}. The probability of rolling a number in that set is 1. You're guaranteed to get one of those results.

Axiom 3: Additivity If two events are mutually exclusive (meaning they cannot happen at the same time), the probability that one or the other occurs is the sum of their individual probabilities.

When rolling a die, the events "rolling a 1" and "rolling a 2" are mutually exclusive. You can't do both on a single roll. Therefore, the probability of rolling a 1 or a 2 is the sum of their probabilities: $P(1) + P(2) = 1/6 + 1/6 = 2/6 = 1/3$.

These three simple rules are all we need to build a powerful framework for thinking about and calculating the chances of complex events.

With these foundational concepts, you're ready to explore how probability works in more complex and interesting scenarios.