Mathematics & Logic

Understanding Probability:
Formulas, Distributions & Applications

Published May 18, 2026 • 10 min read

Probability theory is the mathematical framework we use to analyze and quantify uncertainty. Whether you are a software developer sizing infrastructure scaling limits, a financial analyst evaluating options markets, or a student solving statistics homework, probability provides the bedrock principles for data-driven predictions.

In this technical guide, we will break down the essential axioms of probability, clarify independent vs. dependent events, explore conditional probability matrices, and demonstrate Bayes' theorem.

1. The Axioms of Classical Probability

Historically defined by Pierre-Simon Laplace and formalized by Andrey Kolmogorov, the probability of an event $A$, denoted $P(A)$, is the ratio of favorable outcomes to the total size of the sample space $S$:

$$P(A) = \frac{n(A)}{n(S)}$$

Where $n(A)$ represents the number of favorable elements in the event set and $n(S)$ represents the total number of equally likely sample outcomes. Kolmogorov established three foundational axioms:

The probability of any event is a non-negative real number: $P(A) \ge 0$.
The probability of the entire sample space occurring is exactly one: $P(S) = 1$.
For any sequence of mutually exclusive events, the probability of their union equals the sum of their individual probabilities.

2. Independent vs. Dependent Events

When assessing multiple consecutive events, we must establish whether the outcome of the first event affects the sample parameters of the next.

A. Independent Events

Two events are independent if the occurrence of event $A$ has zero impact on the probability of event $B$ occurring. The joint probability of both events occurring is the product of their individual probabilities:

$$P(A \cap B) = P(A) \times P(B)$$

B. Dependent Events (Conditional Probability)

If the occurrence of event $A$ alters the sample space, the events are dependent. To calculate the joint probability, we utilize the conditional probability of $B$ given that $A$ has already occurred, denoted $P(B|A)$:

$$P(A \cap B) = P(A) \times P(B|A)$$

3. Bayes' Theorem: Updating Prior Beliefs

Bayes' theorem is one of the most powerful formulations in mathematics. It provides a way to update the probability of a hypothesis ($H$) based on new evidentiary data ($E$):

$$P(H|E) = \frac{P(E|H) \cdot P(H)}{P(E)}$$

Where:

$P(H|E)$ represents the posterior probability: the probability of hypothesis $H$ after seeing evidence $E$.
$P(E|H)$ represents the likelihood: the probability of observing evidence $E$ given hypothesis $H$ is true.
$P(H)$ represents the prior probability: our initial assessment of the likelihood of hypothesis $H$.
$P(E)$ represents the marginal likelihood: the total probability of the evidence occurring across all possible hypotheses.

4. Real-World Applications

From algorithm designs to medical statistics, probability guides our most critical calculations:

Machine Learning Spam Filters

Naïve Bayes algorithms scan words inside emails, calculating the posterior probability that a message is "spam" based on the historic occurrences of specific words.

Medical Diagnostics

When a patient tests positive for a rare disease, physicians apply Bayes' theorem to check the true posterior probability of having the disease, adjusting for test false-positive rates.

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