Life is full of uncertainty. Will it rain tomorrow? Will your favorite team win the championship? Will you catch the bus? We constantly make guesses and judgments about the future. Probability gives us a mathematical way to measure this uncertainty.

Think of probability as a scale from 0 to 1. An event with a probability of 0 is impossible—it will never happen. An event with a probability of 1 is certain—it will definitely happen. Everything else falls somewhere in between. A probability of 0.5 means an event has a 50/50 chance of occurring, just like a coin flip.

To calculate a probability, we first need to understand all the possible things that could happen. This complete set of all potential outcomes is called the sample space. For a single coin flip, the sample space is simple: {Heads, Tails}. For a standard six-sided die, the sample space is {1, 2, 3, 4, 5, 6}.

Once we have our sample space, we can define an event. An event is the specific outcome or group of outcomes we're interested in. It's a subset of the sample space. For example, if we roll a die, the event of "rolling an even number" corresponds to the outcomes {2, 4, 6}.

When all outcomes in a sample space are equally likely, we can find the probability of an event with a simple formula. We just divide the number of outcomes in the event by the total number of outcomes in the sample space.

For our dice roll example, the probability of rolling an even number is: $P(\text{Even}) = \frac{3}{6} = \frac{1}{2}$ or 0.5.

Probability isn't arbitrary; it follows three fundamental rules, known as axioms. These simple rules are the bedrock of all probability theory.

This also means, combined with the next rule, that the probability of any event $E$ must be between 0 and 1.

This makes intuitive sense. When you roll a die, it's a certainty that you will get one of the numbers from 1 to 6. The probability of getting a number that is in the sample space is $6/6 = 1$.

For example, you can't roll a 1 and a 3 on a single die at the same time. These events are mutually exclusive. So, the probability of rolling a 1 or a 3 is found by adding their probabilities together.

$P(\text{roll is 1 or 3}) = P(1) + P(3) = \frac{1}{6} + \frac{1}{6} = \frac{2}{6} = \frac{1}{3}$

These basic concepts—sample spaces, events, and the three axioms—form the foundation for understanding how to work with uncertainty in a logical and consistent way.