Probability Calculator

Combinatorics · conditional · Bayes · unions

P(A) = favorable / total

Result

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Enter values and click Calculate

Compute probability using basic rules, conditional probability, or Bayes’ theorem.

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Probability Fundamentals

Probability measures the likelihood of an event occurring, expressed as a number between 0 (impossible) and 1 (certain). It is the foundation of statistics, machine learning, and decision-making.

Basic Probability:  P(A) = Favorable Outcomes / Total Outcomes
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Complement Rule

P(¬A) = 1 − P(A). The probability an event does not happen.

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Addition Rule (OR)

P(A ∪ B) = P(A) + P(B) − P(A ∩ B). Subtract the overlap.

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Multiplication Rule (AND)

P(A ∩ B) = P(A) × P(B|A). For independent events: P(A) × P(B).

Conditional Probability & Bayes’ Theorem

Conditional:  P(A|B) = P(A ∩ B) / P(B)
Bayes’ Theorem:  P(A|B) = P(B|A) × P(A) / [P(B|A) × P(A) + P(B|¬A) × P(¬A)]
Medical Test Example: A disease affects 1% of the population. A test has 95% sensitivity and 5% false positive rate. If you test positive, what is the probability you have the disease?
P(Disease) = 0.01, P(+|Disease) = 0.95, P(+|Healthy) = 0.05
P(+) = 0.95 × 0.01 + 0.05 × 0.99 = 0.059
P(Disease|+) = (0.95 × 0.01) / 0.059 = 0.161 (16.1%)
Despite a “95% accurate” test, there is only a 16% chance of having the disease!

Independent vs. Mutually Exclusive Events

PropertyIndependent EventsMutually Exclusive Events
DefinitionOne event does not affect the otherEvents cannot both occur
P(A ∩ B)P(A) × P(B)0
P(A ∪ B)P(A) + P(B) − P(A)P(B)P(A) + P(B)
ExampleCoin flip and die rollDrawing a red or blue ball
Can both occur?YesNo

Real-World Applications

Medical Diagnosis

Bayes’ theorem updates disease probability after test results, accounting for base rates and test accuracy.

Spam Filtering

Naïve Bayes classifiers compute the probability an email is spam given the words it contains.

Insurance & Risk

Actuaries use conditional probabilities to assess risk and calculate premium rates.

Games & Sports

Poker odds, win probability given the current game state, expected value of bets.

Frequently Asked Questions

Independent events do not affect each other, so P(A ∩ B) = P(A) × P(B). Mutually exclusive events cannot both occur, so P(A ∩ B) = 0. These are different concepts: two events can be independent but not mutually exclusive, and vice versa.
Bayes’ theorem updates a prior probability P(A) based on new evidence B. The posterior P(A|B) = P(B|A) × P(A) / P(B). It is widely used in medical testing, spam filtering, and machine learning.
Conditional probability P(A|B) is the probability of event A occurring given that B has already occurred. The formula is P(A ∩ B) / P(B). It narrows the sample space to only outcomes where B is true.
Use the addition rule: P(A ∪ B) = P(A) + P(B) − P(A ∩ B). For mutually exclusive events, P(A ∩ B) = 0, so it simplifies to P(A) + P(B). Always subtract the overlap to avoid double-counting.
Probability is favorable outcomes divided by total outcomes, ranging from 0 to 1. Odds are favorable to unfavorable (e.g., 3:2). Convert odds to probability: p = favorable / (favorable + unfavorable).
When a disease is rare (low prior), even a highly accurate test produces many false positives from the large healthy population. This is the base rate fallacy. Bayes’ theorem correctly accounts for it, showing the posterior can be much lower than expected.

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