ganak.appProbability Calculator
Free probability calculator with 4 modes: single event, two events (joint, union, conditional, XOR), repeated trials, and Bayes' theorem. See step-by-step solutions with interactive Venn diagrams. Calculate P(A∩B), P(A∪B), P(A|B), complement, and series probability instantly.
Enter probabilities for events A and B
Venn Diagram
Hover results to highlight regions
All Probabilities
Joint, union, exclusive, and conditional results
Complements
The probability each event does not occur
Step-by-Step Solution
Formula, substitution, and result for each step
Given probabilities (independent events)
P(A), P(B)
P(A) = 0.6, P(B) = 0.3
Calculate intersection (multiplication rule)
P(A∩B) = P(A) × P(B)
P(A∩B) = 0.6 × 0.3
P(A∩B) = 0.18
Calculate union (addition rule)
P(A∪B) = P(A) + P(B) − P(A∩B)
P(A∪B) = 0.6 + 0.3 − 0.18
P(A∪B) = 0.72
Calculate exclusive OR
P(A XOR B) = P(A) + P(B) − 2×P(A∩B)
P(A XOR B) = 0.6 + 0.3 − 2×0.18
P(A XOR B) = 0.54
Calculate neither event
P(neither) = 1 − P(A∪B) = P((A∪B)')
P(neither) = 1 − 0.72
P(neither) = 0.28
Conditional probability P(A|B)
P(A|B) = P(A∩B) / P(B)
P(A|B) = 0.18 / 0.3
P(A|B) = 0.6
Conditional probability P(B|A)
P(B|A) = P(A∩B) / P(A)
P(B|A) = 0.18 / 0.6
P(B|A) = 0.3
How to Calculate Probability
Fundamental probability rules explained step by step
Probability measures the likelihood of an event occurring, expressed as a number between 0 (impossible) and 1 (certain). The basic formula is P(A) = favorable outcomes / total outcomes. For example, the probability of rolling a 3 on a fair die is 1/6 because there is one favorable outcome out of six possible outcomes.
Key Probability Formulas
P(A′) = 1 − P(A)
P(A∩B) = P(A) × P(B) (independent)
P(A∪B) = P(A) + P(B) − P(A∩B)
P(A|B) = P(A∩B) / P(B)
The complement rule states that the probability of an event not occurring equals 1 minus the probability it does occur. This is often the easiest way to solve “at least one” problems: calculate the probability of zero occurrences and subtract from 1.
Probability Rules & Formulas
Essential rules for combining probabilities
Multiplication Rule
P(A∩B) = P(A) × P(B|A)
For independent events, simplifies to P(A) × P(B)
Addition Rule
P(A∪B) = P(A) + P(B) − P(A∩B)
For mutually exclusive events, P(A∩B) = 0
Bayes' Theorem
P(A|B) = P(B|A)×P(A) / P(B)
Updates prior belief based on new evidence
Binomial Probability
P(X=k) = C(n,k) × pᵏ × (1−p)ⁿ⁻ᵏ
Exactly k successes in n independent trials
Types of Probability
Understanding theoretical, experimental, and subjective probability
Theoretical Probability
Based on equally likely outcomes using mathematical reasoning. Example: P(heads) = 1/2 for a fair coin. Used when all outcomes are known and equally probable.
Experimental Probability
Based on actual observations and data. Calculated as successes / total trials. As the number of trials increases, experimental probability approaches theoretical probability (Law of Large Numbers).
Conditional Probability
The probability of an event given that another event has occurred. Written P(A|B). Changes the sample space to only outcomes where B is true. Key to understanding dependent events.
Subjective Probability
Based on personal judgment, experience, or expert opinion rather than calculation. Used in risk assessment, weather forecasting, and business decisions where exact probabilities are unknown.
Real-World Applications
Where probability calculations matter in everyday life
Medical Testing
Bayes' theorem calculates the true probability of having a disease given a positive test. A 99% accurate test can still yield mostly false positives if the disease is rare (low prior probability).
Quality Control
Manufacturing uses binomial probability to determine defect rates. If 2% of items are defective, probability helps answer: “What is the chance of finding 3 or more defective items in a batch of 100?”
Games & Gambling
Probability determines expected outcomes in card games, dice games, and lotteries. Understanding house edge, pot odds, and expected value helps make informed decisions.
Risk Assessment
Insurance, finance, and engineering use probability to quantify risk. Series probability answers: “If a component has a 0.1% daily failure rate, what is the probability it fails within a year?”
Frequently Asked Questions
Common questions about probability calculations