Probability is all about measuring uncertainty. To do this in a structured way, we use a concept called a probability space. Think of it as the official rulebook and playing field for any random phenomenon, whether it's flipping a coin, rolling a die, or predicting the weather.

A probability space has three key parts.

In this example, the sample space Ω includes all six possible outcomes of a die roll. The event we're interested in is the set {2, 4, 6}. Since each outcome has a 1/6 chance, the probability of this event is the sum of the probabilities of its individual outcomes: $1/6 + 1/6 + 1/6 = 3/6 = 1/2$.

Outcomes aren't always numbers. A coin flip gives you "heads" or "tails." But to do math, we need numbers. This is where random variables come in. A random variable is a rule that assigns a numerical value to each outcome in the sample space.

Imagine you flip two coins. The sample space is {HH, HT, TH, TT}. We can define a random variable, let's call it $X$, to be the number of heads you get. Here’s how $X$ would map each outcome to a number:

By using the random variable $X$, we can now ask numerical questions, like "What is the probability that $X=1$?” The outcomes that give $X=1$ are HT and TH. If each of the four outcomes is equally likely (with probability 1/4), then the probability that $X=1$ is $1/4 + 1/4 = 1/2$.

A probability distribution tells us the probability for each possible value of a random variable. It's a complete description of how the total probability (which is always 1) is spread out, or distributed, among all the possibilities.

For our two-coin-flip example with the random variable $X$ (number of heads), the probability distribution looks like this:

Notice that the probabilities add up to 1 ($0.25 + 0.50 + 0.25 = 1$). This must always be true for any probability distribution. The distribution can be represented as a table, a graph, or a formula.

To work with probabilities, we need a few basic rules. These laws help us combine probabilities of different events in a logical way.

The sum of the probabilities of all possible outcomes in the sample space must equal 1.

For example, when rolling a die, the events "rolling a 1" and "rolling a 6" are mutually exclusive. The probability of rolling a 1 or a 6 is $P(1) + P(6) = 1/6 + 1/6 = 2/6 = 1/3$.

For instance, what's the probability of flipping a coin and getting heads ($P(A) = 1/2$) AND rolling a die and getting a 6 ($P(B) = 1/6$)? Since these events are independent, the probability is $1/2 \times 1/6 = 1/12$.

These foundational ideas—probability spaces, random variables, distributions, and the basic laws—are the building blocks for understanding more complex models of uncertainty.