Probability Calculator
Compute basic, complementary, joint, conditional and independent probabilities. Enter probabilities (decimals, percentages, or fractions) or raw counts and get step-by-step explanations.
Probability Calculator — a practical guide to basic, conditional, joint and independent probabilities
Probability measures the likelihood of events — a number between 0 (impossible) and 1 (certain). Whether you’re estimating the chance of drawing an ace from a deck, modelling risk, or calculating conditional probabilities in data science, a reliable calculator and clear understanding of the formulas save time and prevent mistakes. This guide blends definitions, practical examples, and step-by-step instructions to use the online Probability Calculator effectively.
Basic probability
The simplest probability is the ratio of favorable outcomes to total possible outcomes when all outcomes are equally likely:
P(A) = (number of favorable outcomes) / (total number of outcomes)
Example: probability of rolling a 4 on a fair six-sided die is 1/6 ≈ 0.1667 = 16.67%.
Complementary probability
The complement of event A is "not A". Its probability is:
P(not A) = 1 − P(A)
Example: if P(rain today) = 0.3, then P(no rain) = 0.7.
Joint probability
Joint probability P(A ∩ B) represents the probability that both A and B occur. If you have raw counts, the joint probability is the joint count divided by total sample size. For independent events, P(A ∩ B) = P(A) × P(B), but that equality does not hold for dependent events.
Conditional probability
Conditional probability P(A|B) is the probability that A occurs given that B has occurred. The formula is:
P(A|B) = P(A ∩ B) / P(B) (provided P(B) > 0)
Conditional probabilities are central in medical testing (sensitivity & specificity), Bayesian inference, and many real-world scenarios where information changes the odds.
Independence vs dependence
Two events A and B are independent if knowledge about B does not change the probability of A:
P(A|B) = P(A) ⇔ P(A ∩ B) = P(A) × P(B)
If P(A ∩ B) differs from P(A)×P(B), the events are dependent and conditional probability must be used.
Bayes’ Theorem (brief)
Bayes’ theorem lets you reverse conditional probabilities:
P(A|B) = P(B|A) × P(A) / P(B)
It’s widely used in diagnostics, machine learning (Naive Bayes), and decision-making under uncertainty.
How to use this calculator
- Select a calculation mode: Basic, Complement, Joint, Conditional, Independent, or Bayes.
- Choose whether you're supplying probabilities directly or raw counts (counts are converted to probabilities using the total).
- Fill relevant fields: P(A), P(B), P(A ∩ B) or counts for A, B, joint counts, and total sample size.
- Check "Show step-by-step" to see formulas and intermediate steps. Click Calculate to get the numeric result and explanation.
- Use Download CSV or Copy Result to save or share the computation details.
Worked examples
1) Basic probability with counts
In a bag with 20 marbles, 5 are red. P(red) = 5/20 = 0.25 = 25%.
2) Joint & conditional from counts
Survey: 100 people; 40 like tea (A), 60 like coffee (B), 25 like both. P(A)=0.4, P(B)=0.6, P(A ∩ B)=0.25. Then P(A|B)=P(A ∩ B)/P(B)=0.25/0.6 ≈ 0.4167 (≈41.67%).
3) Independence check
From same survey: P(A)×P(B)=0.4×0.6=0.24. Since P(A ∩ B)=0.25 ≠ 0.24, A and B are slightly dependent.
Tips and common pitfalls
- Always check the denominator: conditional probability divides by P(B), so P(B) must be > 0.
- Be consistent with formats (decimals vs percentages). This calculator accepts mixed formats and normalizes them.
- When using counts, ensure the joint count does not exceed the margins (joint ≤ min(countA,countB)).
- Remember that independence is an assumption — verify with data rather than presuming it.
Applications
Probability is everywhere: games (dice, cards), risk analysis (finance, insurance), diagnostics (medical tests), machine learning (classification probabilities), and reliability engineering. Understanding conditional probability and independence is critical in modeling real-world phenomena accurately.
Conclusion
This Probability Calculator helps you compute and understand basic, joint, conditional and independent probabilities; it accepts counts or probabilities and shows step-by-step reasoning. For advanced statistical modeling and estimation, complement this tool with statistical software and consider confidence intervals, sampling variability, and model assumptions.
Frequently Asked Questions
Decimals (0.25), percentages (25%), fractions (1/4) and integer counts. The calculator normalizes input to probabilities.
Use mode "Conditional" and either provide P(A ∩ B) and P(B) or joint and marginal counts plus total to compute P(A|B)=P(A ∩ B)/P(B).
The tool compares P(A ∩ B) with P(A)×P(B) and reports whether they are equal within a small numeric tolerance.
Yes — provide joint count (A and B) along with counts for A, B and the total; the calculator will compute required probabilities.
Select "Bayes" mode and enter prior probabilities and likelihoods; the tool computes P(A|B) using Bayes' formula and shows steps.
Results use standard JavaScript numeric precision. When you supply integer counts, fraction computations are exact until converted to decimal representation.
Yes — use Download CSV to save inputs, results and step logs, or Copy Result to copy a short summary to your clipboard.
The conditional probability is undefined. The calculator will warn you and not compute P(A|B) if P(B) equals zero.
This tool supports basic joint and conditional calculations. For complex multi-event models consider using probability distributions and statistical tools such as Bayesian networks or probabilistic programming frameworks.
It’s useful for simple probability calculations and sanity checks. For model training and inference use dedicated ML libraries that handle estimation, uncertainty and model validation.