Probability Calculator
Free probability calculator. Single event, two events (union/intersection), conditional probability
Why This Statistical Analysis Matters
Why: Statistical calculator for analysis.
How: Enter inputs and compute results.
Single Event, Two Events, Conditional (Bayes), Repeated Trial Probabilities
From Fermat & Pascal (1654) to modern AI — probability underpins machine learning, medical diagnosis, finance, and everyday decisions.
Probability Mode
Real-World Scenarios — Click to Load
Inputs
Probability Distribution
Calculation Breakdown
⚠️For educational and informational purposes only. Verify with a qualified professional.
Key Takeaways
- • Probability ranges from 0 (impossible) to 1 (certain)
- • Addition rule for "or": P(A∪B) = P(A) + P(B) - P(A∩B)
- • Multiplication rule for "and" (independent): P(A∩B) = P(A) × P(B)
- • Bayes' theorem reverses conditional probability: P(A|B) = P(B|A)P(A)/P(B)
- • At least once in n trials: P = 1 - (1-p)^n
Did You Know?
How It Works
1. The Sample Space
All possible outcomes of an experiment. For a fair die, the sample space is 1, 2, 3, 4, 5, 6. Probability = favorable outcomes / total outcomes.
2. Classical Probability
P(A) = (number of favorable outcomes) / (total outcomes). Assumes equally likely outcomes. Example: P(rolling 6) = 1/6.
3. Addition Rule
When events can overlap: P(A or B) = P(A) + P(B) - P(A and B). Without the subtraction, you would double-count the overlap.
4. Multiplication Rule
Independent events: P(A and B) = P(A) × P(B). Dependent events require conditional probability: P(A and B) = P(A) × P(B|A).
5. Bayes' Theorem
Reverses conditional probability: P(A|B) = P(B|A)P(A)/P(B). Essential for medical diagnosis, spam filtering, and machine learning.
Expert Tips
Use the Complement Rule
P(at least one) = 1 - P(none) is easier than computing P(1) + P(2) + ...
Draw a Tree Diagram
For conditional probability, tree diagrams make Bayes' theorem intuitive.
Test Independence First
P(A∩B) = P(A)×P(B) only holds for independent events — don't assume it.
Bayes in Medicine
A positive test with 99% sensitivity and 1% disease prevalence means only ~50% true positive rate.
This Calculator vs. Other Tools
| Feature | This Calculator | Wolfram Alpha | TI-84 | Manual |
|---|---|---|---|---|
| Multiple modes | ✅ | ⚠️ | ❌ | ❌ |
| Visual charts | ✅ | ⚠️ | ❌ | ❌ |
| Step-by-step | ✅ | ⚠️ | ❌ | ❌ |
| Bayes calculation | ✅ | ✅ | ❌ | ⚠️ |
| Copy & share | ✅ | ❌ | ❌ | ❌ |
| AI analysis | ✅ | ❌ | ❌ | ❌ |
| No software needed | ✅ | ⚠️ | ❌ | ✅ |
Frequently Asked Questions
What is the difference between independent and mutually exclusive events?
Independent: P(A∩B) = P(A)×P(B). One event doesn't affect the other. Mutually exclusive: P(A∩B) = 0 — they cannot both occur.
When do I add probabilities vs multiply them?
Add for 'or' (union): P(A or B). Multiply for 'and' (intersection): P(A and B) when independent. For overlapping 'or', use P(A) + P(B) - P(A∩B).
How does Bayes' theorem work intuitively?
Bayes reverses conditional probability. Given P(positive test | disease), we find P(disease | positive test). It updates prior belief (prevalence) with new evidence (test result).
What is the gambler's fallacy?
The mistaken belief that past outcomes affect future ones. After 10 heads, the next flip is still 50% heads — each trial is independent.
How do I calculate P(at least one) in repeated trials?
Use the complement: P(at least one) = 1 - P(none) = 1 - (1-p)^n. Much easier than summing P(1) + P(2) + ... + P(n).
What is conditional probability and when do I use it?
P(A|B) = probability of A given B occurred. Use when one event affects another: medical tests, spam filters, quality control.
Can probability be greater than 1?
No. Probability is always between 0 and 1 (or 0% and 100%). If your calculation exceeds 1, check for overlapping events or input errors.
How is probability used in machine learning?
ML models output probabilities: classification confidence, language model next-token probabilities, reinforcement learning policies. Naive Bayes, logistic regression, and neural networks all use probability.
Probability by the Numbers
Official Data Sources
Disclaimer: This calculator provides accurate probability computations using well-established mathematical formulas and axioms. For critical applications in medical diagnosis, legal proceedings, or financial risk assessment, consult a qualified statistician.
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