When we hear the word “probability,” we often think of phrases like “the probability is 50%.” But what does probability actually mean? Historically, two major approaches have shaped our understanding: Laplace’s classical probability and Kolmogorov’s modern probability.

In this article, we’ll gently walk through both definitions and set the stage for future discussions about probability distributions and probability density functions.

Laplace’s Probability (Classical Definition)

In the 18th century, Pierre-Simon Laplace defined probability as:

Probability = Number of favorable outcomes ÷ Total number of possible outcomes

This works well for experiments like dice or coin tosses, where all outcomes are finite and symmetric.

Example: Rolling a die

If we roll a six-sided die once, the probability of getting an odd number (1, 3, 5) is:

This relies on the assumption of symmetry: every outcome is equally likely.

Limitations

Laplace’s definition has important limitations:

• It does not apply when there are infinitely many outcomes (e.g., choosing a real number)
• It struggles when outcomes are biased (e.g., a weighted coin)

To overcome these issues, we turn to Kolmogorov’s probability.

Kolmogorov’s Probability (Modern Definition)

In 1933, the Soviet mathematician Andrey Kolmogorov introduced an axiomatic system for probability.

Probability Space $(\Omega, \mathcal{F}, P)$

Kolmogorov formalized probability with three components:

1. $\Omega$: the set of all possible outcomes (sample space)
2. $\mathcal{F}$: a collection of subsets of $\Omega$ called events (a σ-algebra)
3. $P$: a probability function that assigns a value in $[0, 1]$ to each event in $\mathcal{F}$

Note: A σ-algebra is, roughly speaking, a collection of subsets that is closed under union, intersection, and complement—even when we take infinitely many of them. It ensures we can handle probabilities consistently without contradictions.

Axioms

The probability function $P$ must satisfy:

1. Non-negativity: $P(A) \geq 0$
2. Normalization: $P(\Omega) = 1$
3. Additivity: For disjoint events $A$ and $B$, $P(A \cup B) = P(A) + P(B)$

This definition no longer depends on “counting outcomes” and works for continuous spaces.

Example: Dropping a needle on [0,1]

Imagine dropping a needle randomly somewhere between 0 and 1 cm on a line. The probability that it lands in the interval $[0, 0.5]$ is:

Here, probability is measured by length (or area, volume, etc.), making continuous outcomes manageable.

Laplace vs Kolmogorov: A Comparison

Feature	Laplace	Kolmogorov
Suitable for	Finite, symmetric cases	Infinite, continuous cases
Definition	Favorable outcomes ÷ Total outcomes	Axiomatic definition
Limitations	Biased or continuous cases	Abstract, but highly general

Let’s learn both types of probability with interactive visual demo!

Laplace vs Kolmogorov: Interactive Probability Demo

Understand the fundamental difference between counting-based and measure-based probability

Laplace (Classical)Kolmogorov (Modern)Comparison

Laplace's Classical Probability

Formula: P(Event) = Favorable Outcomes ÷ Total Outcomes

Example: What's the probability of rolling an odd number with a fair six-sided die?

Key Points:

• All outcomes must be equally likely (symmetry assumption)
• Works well for finite discrete cases
• Simply count favorable vs. total outcomes