Life is full of uncertainty. Will it rain tomorrow? Will your favorite team win the championship? Will the bus be on time? We can't know for sure, but we can often make a pretty good guess. Probability is the mathematics of guessing. It's a way to measure how likely something is to happen.

Think about flipping a coin. There are two possible results: heads or tails. Assuming the coin is fair, each outcome is equally likely. We'd say there's a 50% chance of getting heads. Probability turns this intuitive idea into a precise number.

This number always falls on a scale from 0 to 1.

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

Everything else falls somewhere in between. An event with a probability of 0.25 is less likely than an event with a probability of 0.8.

To calculate a probability, we first need to understand all the possible outcomes of a situation. This complete set of all possible results is called the sample space.

For a coin flip, the sample space is simple: {Heads, Tails}. For a single roll of a six-sided die, the sample space is {1, 2, 3, 4, 5, 6}. It’s a list of everything that could possibly happen.

An event is a specific outcome, or a collection of outcomes, that we are interested in. It's a subset of the sample space. Using the two-dice example, an event could be:

• "Rolling doubles."
• "The sum of the dice is 3."
• "The first die is a 4."

Let's consider the event of "rolling a sum of 7." The specific outcomes from our sample space that make up this event are: (1, 6), (2, 5), (3, 4), (4, 3), (5, 2), and (6, 1). There are 6 favorable outcomes.

Once we have the sample space and the event, we can often calculate the probability with a simple formula, as long as all outcomes are equally likely.

Let's apply this. The probability of rolling a 7 with two dice is:

$P(\text{Sum is 7}) = \frac{6}{36} = \frac{1}{6}$

There are 6 ways to get a 7, and 36 total possible outcomes. So the probability is about 0.167, or 16.7%.

Probability isn't arbitrary; it follows three fundamental rules, often called axioms. These rules ensure that our calculations are consistent and make logical sense.

From these axioms, we get other useful properties. One of the most important is the complement rule. If the probability of an event happening is $P(A)$, then the probability of it not happening is $1 - P(A)$.

For instance, we calculated the probability of rolling a 7 is $\frac{1}{6}$. The probability of not rolling a 7 is therefore $1 - \frac{1}{6} = \frac{5}{6}$.

Let's test your understanding of these core concepts.