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Probability Calculator

Calculate probabilities for single and multiple events, conditional probability, Bayes' theorem, and more with detailed step‑by‑step solutions.

How many times the event occurs (for at‑least‑once probability)

Probability Formulas

Single Event: P(A) = Favorable / Total
Independent: P(A and B) = P(A)·P(B)
P(A or B) = P(A) + P(B) - P(A and B)
Conditional: P(A|B) = P(A and B) / P(B)
Bayes: P(A|B) = P(B|A)·P(A) / P(B)
Complement: P(not A) = 1 - P(A)

Mastering Probability: The Complete Guide

Probability quantifies the likelihood of an event occurring, ranging from 0 (impossible) to 1 (certain). This Probability Calculator provides four essential tools: single event probability, multiple event combinations (AND/OR for independent, mutually exclusive, and dependent events), conditional probability, and Bayes' theorem. Each calculation includes a step‑by‑step derivation, visual probability bar, and conversion to percentage and odds. Whether you're studying statistics, analyzing risks, or making data‑driven decisions, this tool delivers clarity and precision.

Single Event Probability

The probability of a single event is the number of favorable outcomes divided by the total number of equally likely outcomes. For example, the probability of rolling a 4 on a fair six‑sided die is 1/6. The calculator also computes the "at least once" probability over multiple trials using the complement rule: P(at least one) = 1 − (1 − P)ⁿ.

Multiple Events: AND, OR, and Relationships

  • Independent Events: Occurrence of one does not affect the other. P(A and B) = P(A) × P(B).
  • Mutually Exclusive Events: Cannot occur together. P(A or B) = P(A) + P(B).
  • Dependent Events: The probability of one affects the other; you must provide P(A and B).

📊 Conditional Probability & Bayes' Theorem

Conditional probability, P(A|B), measures the probability of A given that B has occurred. Bayes' theorem extends this to update prior beliefs with new evidence: P(A|B) = P(B|A)·P(A) / P(B). It is foundational in medical diagnosis, spam filtering, machine learning, and forensic science.

Real‑World Applications

  • Games of Chance: Dice, cards, lotteries.
  • Risk Assessment: Insurance, finance, project management.
  • Quality Control: Defect rates and sampling.
  • Medical Testing: Interpreting test accuracy with Bayes' theorem.
  • Machine Learning: Naive Bayes classifiers and probabilistic models.

Frequently Asked Questions

What is the difference between probability and odds?

Probability = favorable/total. Odds = favorable:unfavorable. Example: 1/6 → 1:5.

What does "mutually exclusive" mean?

Events that cannot occur together. P(A and B)=0; P(A or B)=P(A)+P(B).

How do I calculate "at least one" probability?

Use complement: P(at least one) = 1 − P(none). Enter trials in Single Event tab.

When should I use Bayes' theorem?

To update probabilities with new evidence—medical diagnosis, spam filtering, quality control.

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