Probability Calculator

Calculate event probabilities, combinations, and statistical likelihood 🎯

Favorable Outcomes:

Total Possible Outcomes:

P(A and B):

P(B):

P(A):

P(B|A):

P(B):

Probability drives decision-making across every field—from medical diagnosis to financial risk assessment, from weather forecasting to artificial intelligence algorithms. Understanding probability calculations enables you to quantify uncertainty, make data-driven predictions, and optimize outcomes in uncertain environments. Whether you’re analyzing experimental results, assessing investment risks, or designing machine learning models, probability mathematics provides the theoretical foundation for reasoning under uncertainty.

🎲 Probability Foundations

Probability theory connects to numerous mathematical concepts. Enhance your probability analysis with our Combination Calculator for counting outcomes and Bayes’ Theorem Calculator for advanced conditional probability scenarios.

Table of Contents

Probability Theory Fundamentals

Probability quantifies the likelihood of events occurring, providing a mathematical framework for handling uncertainty and making informed decisions when outcomes cannot be predicted with certainty.

Basic Probability Principles

Classical probability calculation relies on equally likely outcomes:

Fundamental Formula

P(Event) = Favorable Outcomes / Total Possible Outcomes

Range: 0 ≤ P(Event) ≤ 1

Conditional Probability

Conditional probability calculates the likelihood of an event given that another event has occurred:

Conditional Formula

P(A|B) = P(A and B) / P(B)

Read as: “Probability of A given B”

Bayes’ Theorem

Bayes’ theorem provides a way to update probabilities based on new evidence:

Bayes’ Formula

P(A|B) = P(B|A) × P(A) / P(B)

Where: P(A) = prior, P(B|A) = likelihood, P(B) = evidence

Frequently Asked Questions

What’s the difference between theoretical and experimental probability?

Theoretical probability is calculated using mathematical principles assuming equally likely outcomes, while experimental probability is determined by actually performing experiments and recording results. As the number of trials increases, experimental probability tends to approach theoretical probability.

When do I use conditional probability?

Use conditional probability when the occurrence of one event affects the probability of another event. Common applications include medical testing (probability of disease given test results), quality control (defect probability given certain conditions), and financial analysis (market movements given economic indicators).

How is Bayes’ theorem used in real applications?

Bayes’ theorem is fundamental to machine learning algorithms, medical diagnosis systems, spam filtering, and risk assessment. It allows updating beliefs or probabilities as new evidence becomes available, making it essential for adaptive decision-making systems.

Can probability be greater than 1 or less than 0?

No, probability values must always be between 0 and 1 inclusive. A probability of 0 means the event cannot occur, while a probability of 1 means the event is certain to occur. Values outside this range indicate calculation errors or misapplied probability rules.

What’s the relationship between probability and statistics?

Probability deals with predicting future outcomes based on known parameters, while statistics involves analyzing past data to infer unknown parameters. Probability provides the theoretical foundation that statistics uses to make inferences about populations from sample data.