Probability is simply a way to measure how likely something is to happen. It's a number between 0 and 1. If the probability of an event is 0, it's impossible. If it's 1, it's a sure thing. Everything else falls somewhere in between.

We often express probability as a fraction or a decimal. The basic idea is captured in a simple formula, where $E$ stands for the event we're interested in:

For a standard six-sided die, the probability of rolling a 4 is $1/6$. There's one way to get a 4, and there are six possible outcomes in total.

While the basic formula is a great start, we don't always have a neat set of equally likely outcomes. Probabilities can come from different sources, which we can group into three main types.

This is the most straightforward type. Think of games of chance: flipping coins, rolling dice, or drawing a card from a well-shuffled deck. We can calculate the probability without ever performing an experiment because we know all the possible outcomes and that each one is equally likely.

Sometimes, we can't reason our way to a probability. Instead, we have to collect data. For example, to find the probability that a random person is left-handed, we'd have to survey a group of people. If we find that 10 out of 100 people are left-handed, our empirical probability is $10/100$, or 0.1. This type of probability comes from what has already happened.

What's the probability that your favorite team will win its next game? There's no deck of cards or historical data to give a perfect answer. This is where subjective probability comes in. It's an educated guess based on knowledge, experience, and intuition. A sports analyst might say a team has a 70% chance of winning based on player health, recent performance, and other factors.

Calculating the probability of a single event is useful, but we often want to know the chances of multiple events happening. Two basic rules help us do this: the addition rule and the multiplication rule.

The Addition Rule (for 'or' events)

The addition rule helps us find the probability that at least one of two events will occur. For example, what's the probability of rolling a 2 or a 3 on a die?

Since these two events can't happen at the same time (they are mutually exclusive), we can simply add their individual probabilities.

For our die roll: $P(\text{2 or 3}) = P(2) + P(3) = 1/6 + 1/6 = 2/6 = 1/3$.

The Multiplication Rule (for 'and' events)

The multiplication rule is for finding the probability that two or more events will both happen. For this to work, the events must be independent—meaning the outcome of one doesn't affect the outcome of the other.

Flipping a coin twice is a classic example. What's the probability of getting heads on the first flip and heads on the second flip? The first flip doesn't change the odds for the second one, so the events are independent. We just multiply their probabilities.

For two heads in a row: $P(\text{heads and heads}) = P(\text{heads}) \times P(\text{heads}) = 1/2 \times 1/2 = 1/4$.

These foundational ideas are the building blocks for understanding risk, making predictions, and navigating a world full of uncertainty. By classifying probabilities and knowing how to combine them, you can start to make sense of the chances all around you.