Probability is simply a way of measuring how likely something is to happen. It's a number between 0 and 1, where 0 means an event is impossible and 1 means it's certain. A 0.5 probability means an event has a 50/50 chance of occurring, like a fair coin landing on heads.

To calculate it, we look at all the possible outcomes of a situation. This complete set of possibilities is called the sample space. An event is the specific outcome or group of outcomes we're interested in.

The basic formula for probability is straightforward:

So, the probability of rolling a 4 is $1/6$, because there is one favorable outcome (rolling a 4) and six total possible outcomes. The probability of rolling an even number is $3/6$, or $1/2$, since there are three favorable outcomes (2, 4, and 6).

A probability distribution is a function or a table that shows the probabilities of every possible outcome in an experiment. It gives you the full picture of what might happen. Distributions can be simple, like for a single die roll, or much more complex.

For our six-sided die, the distribution is uniform because each outcome has an equal chance of occurring.

Distributions come in two main flavors. Discrete distributions, like the die roll, have a finite number of possible outcomes. Continuous distributions deal with outcomes that can take any value within a range, like the height of a person or the temperature tomorrow.

The expected value is the long-run average outcome of an experiment if you were to repeat it many times. It's a weighted average, where each possible outcome is weighted by its probability. It's what you can 'expect' to happen on average over time, even if it's not a possible outcome in a single trial.

The formula looks like this, where $x$ is an outcome and $P(x)$ is its probability:

Let's imagine a simple game. You pay $1 to play. If you roll a 6 on a fair die, you win $5. If you roll anything else, you win nothing. Your net outcomes are a gain of $4 ($5 prize - $1 cost) or a loss of $1. What's the expected value of playing this game?

The probability of winning is $1/6$, and the probability of losing is $5/6$. So, the expected value is:

On average, you can expect to lose about 17 cents every time you play. This tells you that, over the long run, the game is not profitable for the player.

Variance measures how spread out a set of outcomes are from their average (the expected value). A low variance means the outcomes tend to be very close to the expected value. A high variance means the outcomes are spread out over a much wider range.

Mathematically, variance is the average of the squared differences from the expected value, $E(X)$. This might sound complicated, but the idea is simple: it quantifies the 'risk' or 'volatility' of the outcomes.

A related and more intuitive measure is the standard deviation, which is just the square root of the variance. It's measured in the same units as the original data, making it easier to interpret.

The Law of Large Numbers is a powerful principle that connects theory to the real world. It states that as you perform an experiment more and more times, the average of the results will get closer and closer to the expected value.

If you flip a coin 10 times, you might get 7 heads and 3 tails, an average of 0.7 heads per flip. This is far from the expected value of 0.5. But if you flip it 10,000 times, you're very likely to get a result extremely close to 5,000 heads, for an average near 0.5.

This law is the foundation for why casinos, insurance companies, and investment strategies can be profitable. While individual events are unpredictable, the average outcome over many events becomes very predictable.

Let's review these foundational concepts.

Understanding these core ideas—probability, distributions, expected value, variance, and the law of large numbers—provides a solid base for analyzing situations involving uncertainty.