Probability starts with a clear framework. An experiment is any process with an uncertain result, like rolling a die. Each possible result is an outcome. The set of all possible outcomes is the sample space, which we usually call $S$. For a single six-sided die, the sample space is simple and finite: $S = \{1, 2, 3, 4, 5, 6\}$.

An event is just a specific outcome or a set of outcomes we're interested in. In set theory terms, an event is any subset of the sample space. For our die roll, an event could be simple, like rolling a 5. Or it could be more complex, like rolling an even number. The event of rolling an even number, let's call it $A$, would be the subset $A = \{2, 4, 6\}$.

Since events are sets, we can use set operations to combine or modify them. This is how we handle more complex scenarios.

• Union (A ∪ B): This is event A or event B (or both) happening. If $A$ is rolling an even number and $B$ is rolling a number greater than 4 ($B=\{5, 6\}$), then $A \cup B = \{2, 4, 5, 6\}$.
• Intersection (A ∩ B): This is event A and event B happening. Using the same events, $A \cap B = \{6\}$. It's the outcome that satisfies both conditions.
• Complement (A'): This is event A not happening. The complement of rolling an even number is rolling an odd number: $A' = \{1, 3, 5\}$.

To move from sets to actual probabilities, we need a few ground rules. These are the axioms of probability—fundamental truths that we accept without proof. They ensure that our calculations are consistent and logical. All of probability theory is built on these three simple ideas.

The concept of mutually exclusive events is key to the third axiom. Two events are mutually exclusive if they cannot happen at the same time. In set terms, their intersection is the empty set ($A \cap B = \emptyset$). For a single die roll, the events "rolling a 1" and "rolling a 6" are mutually exclusive. You can't do both in one roll.

This leads to our first tool for calculation. If we have a set of simple, mutually exclusive outcomes, each with an equal chance of occurring, we can find the probability of any event $A$ by counting.

Let's go back to our die. The sample space $S$ has 6 equally likely outcomes. What is the probability of our event $A$, rolling an even number? The event $A$ includes the outcomes {2, 4, 6}, so there are 3 outcomes in $A$.

Using our formula, $P(A) = 3/6 = 0.5$. This simple calculation works because it respects all three axioms. Each outcome has a probability of 1/6 (non-negative), the sum of all six is 1, and since each outcome is mutually exclusive, the probability of the event is the sum of the probabilities of its outcomes ($1/6 + 1/6 + 1/6 = 3/6$).

Time to check your understanding of these core concepts.

With these foundations—sample spaces, events as sets, and the axioms—we have a solid framework for tackling more complex probability problems.