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🎲 Probability Calculator

Compute union, intersection, conditional probability, and complement for independent and dependent events.

Event Likelihood — AND, OR, and Conditional Rules

BrainyCalculators editorial insight — unique to this tool

Probability assigns 0–1 to event likelihood — a fair coin P(heads) = 0.5; rolling at least one six in two dice uses complement: 1 − (5/6)² ≈ 0.306. Insurance actuaries combine independent risks; quality teams compute P(defect) from historical batches. Conditional probability P(A|B) updates beliefs when new evidence arrives — medical test accuracy depends on base rate, not sensitivity alone.

When to use this calculator

Use for chance of events with known or estimated probabilities. For counting equally likely outcomes (before dividing), start with Combination or Permutation.

Reference Value Context
Fair die, one six 1/6 ≈ 0.167 Single trial
Two dice, sum 7 6/36 = 1/6 Most common sum
Independent AND P(A)×P(B) Multiply
At least one 1 − P(none) Complement trick

Not what you need? For counting arrangements, use Permutation/Combination first. For survey margin of error, use Confidence Interval or Sample Size.

Counting possible outcomes in a lottery or committee?

This page applies probability rules to events. For nCr and nPr counting, use the Combination Calculator →

What is Probability?

A probability calculator applies event rules: complements, unions, intersections, and conditional formulas P(A|B). It works from stated probabilities or simple event models.

Use this page for “what is the chance that…” logic. For counting how many outcomes exist in a sample space, use Combination or Permutation calculators.

For standardizing a sample mean with a z-score, use the Z-Score Calculator instead.

Key Probability Formulas

Basic Probability
P(A) = Favorable outcomes / Total outcomes
Complement Rule
P(not A) = 1 − P(A)
Multiplication Rule (Independent)
P(A and B) = P(A) × P(B)
Addition Rule (General)
P(A or B) = P(A) + P(B) − P(A and B)
Conditional Probability
P(A|B) = P(A ∩ B) / P(B)

How to Use This Calculator

  1. 1
    Choose a Calculation Type
    Select Single Event for basic probability, Two Events to combine probabilities with AND/OR, or Conditional for P(A|B).
  2. 2
    Enter Your Values
    For single events, enter favorable and total outcomes. For two events, enter P(A) and P(B) as decimals between 0 and 1.
  3. 3
    Select Event Relationship
    For two events, choose Independent (events do not affect each other) or Mutually Exclusive (events cannot both occur).
  4. 4
    Read the Results
    Results show probability as a decimal, percentage, and fraction. The complement (NOT event) is always included.

Worked Example

Rolling a standard six-sided die — what is the probability of getting an even number?

Even numbers on a die: {2, 4, 6} → Favorable = 3
Total outcomes = 6
P(even) = 3 / 6 = 0.5 = 50%
P(not even) = 1 − 0.5 = 0.5 = 50%

How the Probability Calculator Works

Formula, assumptions, and calculation steps for this statistics tool.

Formula Used

P(A or B) = P(A) + P(B) - P(A and B)

Methodology

Statistics calculators organize sample data, apply the selected descriptive or inferential formula, and report the statistic with interpretation.

Calculation Steps

  1. Enter raw values or summary statistics.
  2. Clean separators and count the sample size.
  3. Apply the relevant statistic, probability, or confidence formula.
  4. Display the result with context such as degrees of freedom, percentile, or strength.

Assumptions and Limits

  • Samples should be representative of the population being studied.
  • Normality or independence assumptions apply only where the selected method requires them.
  • Rounded results may differ slightly from spreadsheet software.

Frequently Asked Questions

Probability is a number between 0 and 1 that measures how likely an event is to occur. A probability of 0 means the event is impossible; 1 means it is certain. It is calculated as the number of favorable outcomes divided by the total number of possible outcomes, assuming all outcomes are equally likely.

Two events are mutually exclusive (or disjoint) if they cannot both occur at the same time. For example, a coin flip cannot be both heads and tails. For mutually exclusive events, P(A and B) = 0 and P(A or B) = P(A) + P(B).

Two events are independent if the occurrence of one does not affect the probability of the other. For example, rolling a die twice — the result of the first roll does not affect the second. For independent events, P(A and B) = P(A) × P(B).

Conditional probability P(A|B) is the probability of event A occurring given that event B has already occurred. It is calculated as P(A|B) = P(A ∩ B) / P(B). This is fundamental to Bayes's theorem and is used widely in statistics, medicine, and machine learning.

Real-World Applications

🏥
Medical Diagnosis & Screening
Bayes' theorem combines disease prevalence with test sensitivity and specificity to calculate the probability that a positive test result indicates actual disease — the positive predictive value. For rare diseases, even a highly accurate test can have a surprising number of false positives.
💹
Financial Risk Modelling
Value at Risk (VaR) and expected shortfall calculations use probability distributions to estimate the likelihood of portfolio losses exceeding a threshold — informing capital reserve requirements and risk limit setting in banks and investment funds.
🎰
Casino & Gambling Analysis
Casino game design uses probability to set house edges precisely — roulette (2.7% edge on European single-zero), blackjack (0.5% with optimal strategy), and slot machines (2–15% RTP shortfall). Understanding these probabilities is essential for evaluating the expected cost of gambling.
🌦️
Weather Forecasting
Weather forecasts express probability explicitly — a "40% chance of rain" means that in historical weather patterns with similar atmospheric conditions, precipitation occurred in 40% of cases. Ensemble forecast models generate probability distributions over possible future weather states.
🔩
Reliability Engineering
Calculate the probability that a system operates without failure over its design life — combining component failure rates and redundancy configurations to determine system-level reliability for aircraft, nuclear plants, medical devices, and other safety-critical systems.
⚖️
Legal & Forensic Statistics
Forensic evidence (DNA matches, fingerprint similarities) is presented as likelihood ratios — the probability of the evidence given guilt divided by the probability given innocence. Misapplication of conditional probability (the "prosecutor's fallacy") has contributed to wrongful convictions.

Common Mistakes

1
The gambler's fallacy — treating independent events as dependent
After flipping heads 10 times in a row, many people believe tails is "due" — but each flip remains a 50/50 event regardless of past outcomes. Past results do not influence future probabilities for truly independent events. The coin has no memory. This fallacy leads to poor decisions in gambling, investing, and any domain involving independent random events.
2
Confusing P(A|B) with P(B|A)
The probability of having cancer given a positive test P(cancer|positive) is not the same as the probability of a positive test given cancer P(positive|cancer). This confusion — known as the base rate fallacy or prosecutor's fallacy — produces dramatically incorrect conclusions. Bayes' theorem is required to correctly calculate the conditional probability of the hypothesis given the evidence.
3
Adding probabilities of non-mutually-exclusive events
The addition rule P(A or B) = P(A) + P(B) only applies when A and B cannot both occur simultaneously. For events that can co-occur, the inclusion-exclusion principle applies: P(A or B) = P(A) + P(B) − P(A and B). Omitting the intersection term causes the probability to be overstated.
4
Multiplying probabilities of dependent events
The multiplication rule P(A and B) = P(A) × P(B) only applies for independent events. If drawing cards from a deck without replacement, the second draw's probability depends on the first — P(A and B) = P(A) × P(B|A) must be used. Assuming independence when dependence exists produces incorrect joint probabilities.
5
Neglecting the base rate in conditional probability
A disease test with 99% accuracy still produces many false positives if the disease is rare (1 in 10,000 prevalence). Of 1,000,000 people tested, ~10,000 false positives occur alongside 99 true positives — the positive predictive value is only ~1%. Ignoring base rate prevalence systematically overestimates the probability that a rare event has occurred.

Probability Rules Quick Reference

Rule Formula Condition
Basic probability P(A) = favourable / total Equally likely outcomes
Complement rule P(not A) = 1 − P(A) Always applies
Addition rule P(A or B) = P(A) + P(B) Mutually exclusive only
General addition P(A or B) = P(A)+P(B)−P(A∩B) Any two events
Multiplication rule P(A and B) = P(A) × P(B) Independent events only
Conditional probability P(A|B) = P(A∩B) / P(B) P(B) > 0

References

  1. Kolmogorov, A.N. Foundations of the Theory of Probability. Chelsea Publishing, 1956.
  2. Feller, W. An Introduction to Probability Theory and Its Applications, Vol. 1. Wiley, 1968.
  3. Gelman, A. et al. Bayesian Data Analysis. CRC Press, 2013.
  4. Kahneman, D. Thinking, Fast and Slow. Farrar, Straus and Giroux, 2011.
  5. Ross, S.M. A First Course in Probability. Pearson, 2019.

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