Probability axioms

The probability axioms are the basic rules that help us understand and work with probability theory. These rules were introduced by a Russian mathematician named Andrey Kolmogorov in 1933. Just like how building blocks fit together to make a structure, these axioms help us make sense of chance and uncertainty in many areas, such as math and science.

These axioms don’t pick a specific way to think about probability, but they make sure that no matter how we look at it, the rules still work. For example, Cox's theorem shows how probability can be understood as a way to measure how likely or believable different ideas are. Meanwhile, Dutch book arguments explain why smart decision-makers should base their choices on measures of probability.

One important rule, called σ-additivity, came from ideas in measure theory created by Lebesgue. Some people prefer a simpler version called finite additivity, which works well for certain practical uses. Together, these axioms give us a strong and reliable framework for studying probability.

Kolmogorov axioms

To understand probability, we use a set of rules called the Kolmogorov axioms, created by a mathematician named Andrey Kolmogorov in 1933. These rules help us make sense of chance and uncertainty in many areas, like games or science.

These axioms need three main things: a list of all possible outcomes, a way to group those outcomes into events, and a way to assign probabilities to each event. Together, these rules create a probability space, which is a special structure that helps us study probabilities in a clear and logical way. The probability of any event is always a number between 0 and 1, and the probability that some outcome will happen is always 1. Also, if you have several events that cannot happen at the same time, the probability of any one of them happening is just the sum of their individual probabilities.

Elementary consequences

The Kolmogorov axioms help us understand how probability works in a way that matches classic ideas about chance. For example, if you have an event A and its opposite (called A complement), the probability that either A happens or its opposite happens must add up to 1. This means the probability of the opposite happening is just 1 minus the probability of A happening.

We also know that the probability of something impossible, like saying no outcome occurs at all, is zero. And if one event is part of another event, the bigger event’s probability can’t be smaller than the smaller one’s. All probabilities must be between 0 and 1, inclusive.

Simple example: Coin toss

Imagine flipping a coin once. It can land on heads (H) or tails (T), but not both. We don’t assume the coin is fair.

According to Kolmogorov's probability rules:

• The chance of getting neither heads nor tails is 0.
• The chance of getting either heads or tails is 1.
• The probabilities of heads and tails always add up to 1.