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 or deciding if we should buy a lottery ticket. It gives us a number between 0 and 1 to represent chance.

A probability of 0 means an event is impossible. The probability that a standard six-sided die will land on the number 7 is 0. A probability of 1 means an event is certain. The probability that the sun will rise tomorrow is, for all practical purposes, 1.

Everything else falls somewhere in between. A coin flip has two possible outcomes, heads or tails. If the coin is fair, each outcome is equally likely. So, the probability of getting heads is 1 out of 2, or 0.5.

To calculate probability, we first need to know all the possible results of an action. This complete set of possible outcomes is called the sample space.

For a single die roll, the sample space is {1, 2, 3, 4, 5, 6}. There are six total possible outcomes.

An event is the specific outcome or group of outcomes we are interested in.

Let's say our event is rolling a 4. There is only one way for this to happen. Using our formula:

$P(\text{rolling a 4}) = 1/6$

What if our event is "rolling an even number"? The favorable outcomes are {2, 4, 6}. There are three of them. So:

$P(\text{rolling an even number}) = 3/6 = 1/2$

The sample space can get larger. If we roll two dice, the total number of outcomes is 36. Each die has 6 possibilities, so we have 6 × 6 = 36 unique combinations.

The graphic above shows all 36 possible outcomes when rolling two dice. The six highlighted outcomes are those where the sum of the dice is 7. So, the probability of rolling a 7 is:

$P(\text{sum is 7}) = 6/36 = 1/6$

Two simple rules can help us combine probabilities: the Addition Rule and the Complement Rule.

The formula is straightforward: $P(A \text{ or } B) = P(A) + P(B)$

What is the probability of rolling a 2 or a 3? $P(\text{2 or 3}) = P(2) + P(3) = 1/6 + 1/6 = 2/6 = 1/3$.

Since an event must either happen or not happen, their probabilities must add up to 1. $P(A) + P(\text{not } A) = 1$

This gives us a handy formula: $P(\text{not } A) = 1 - P(A)$

For example, what is the probability of not rolling a 6? We could add the probabilities of rolling a 1, 2, 3, 4, and 5. Or, we can use the complement rule:

$P(\text{not 6}) = 1 - P(6) = 1 - 1/6 = 5/6$

This rule is incredibly useful when dealing with more complex problems where calculating the probability of an event directly is much harder than calculating its complement.

Let's review these core ideas before moving on.

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These building blocks—sample spaces, events, and the basic rules—are the foundation of probability. Mastering them allows us to understand and analyze the random world around us.