Probability is a way of measuring uncertainty. It tells us how likely an event is to happen, assigning it a number between 0 and 1. A probability of 0 means the event is impossible, while a probability of 1 means it's absolutely certain. Most of life happens somewhere in between.

Think about flipping a standard coin. There are two possible outcomes: heads or tails. Assuming the coin is fair, each outcome is equally likely. So, the chance of getting heads is one out of two, or 0.5. The same goes for tails. This simple idea is the foundation of probability.

To calculate probabilities, we first need to understand what's possible. The set of all possible outcomes of an experiment is called the sample space. For a coin flip, the sample space is {Heads, Tails}. For a single roll of a standard six-sided die, the sample space is {1, 2, 3, 4, 5, 6}.

An event is any specific outcome or group of outcomes we're interested in. It's a subset of the sample space. Using our die example, 'rolling a 3' is an event. 'Rolling an even number' is also an event, which corresponds to the set of outcomes {2, 4, 6}.

The probability of a simple event is the ratio of the number of favorable outcomes to the total number of possible outcomes.

So, the probability of rolling an even number on a die is calculated by counting the favorable outcomes ({2, 4, 6}, which is 3 outcomes) and dividing by the total number of outcomes (6). The probability is $3/6 = 1/2$ or $0.5$.

Sometimes we want to know the probability of combined events, like the chance of one event or another occurring. This is where the addition rule comes in.

For events that can't happen at the same time, known as mutually exclusive events, we simply add their probabilities. For instance, you can't roll a 2 and a 4 on a single die roll. The probability of rolling a 2 or a 4 is the sum of their individual probabilities.

The probability of rolling a 2 is $1/6$, and the probability of rolling a 4 is $1/6$. So, $P(\text{2 or 4}) = 1/6 + 1/6 = 2/6 = 1/3$.

What about the probability of two events happening one after another? For that, we use the multiplication rule. This rule applies to independent events—where the outcome of one event doesn't affect the outcome of the other. Flipping a coin twice is a classic example. The result of the first flip has no impact on the second.

The probability of getting heads on one flip is $1/2$. To find the probability of getting heads twice in a row, we multiply: $P(\text{Heads and Heads}) = 1/2 \times 1/2 = 1/4$.

So far, we've discussed theoretical probability. This is what we expect to happen based on a perfect model. We reason that a coin has a 50% chance of landing on heads because there are two equally likely sides. We don't actually need to flip the coin to come to this conclusion.

Empirical probability, on the other hand, is based on observation and data from real experiments. It's calculated by running an experiment many times and counting the number of times the event of interest occurs.

If you flip a coin 100 times and it lands on heads 53 times, the empirical probability of getting heads is $53/100$, or $0.53$. This isn't exactly the theoretical probability of $0.5$, but it's close.

This highlights an important idea called the law of large numbers. It states that as you repeat an experiment a large number of times, the empirical probability will get closer and closer to the theoretical probability. Your first 10 flips might give you 7 heads (an empirical probability of 0.7), but after 10,000 flips, the result will be much nearer to the expected 0.5.

These core concepts—sample spaces, events, and the basic rules of addition and multiplication—are the building blocks for understanding a world full of uncertainty.