P(A∩B) — Joint Probability
Independent: P(A∩B)=P(A)P(B). Dependent: P(A∩B)=P(A)P(B|A). Joint tables, marginals, Venn.
Why This Statistical Analysis Matters
Why: Joint probability underpins conditional probability, Bayes, and contingency analysis.
How: Enter P(A), P(B) or joint table. Get P(A∩B), P(A∪B), P(A|B), P(B|A).
- ●P(A∩B)=P(A)P(B) if independent
- ●P(A∪B)=P(A)+P(B)−P(A∩B)
- ●P(A|B)=P(A∩B)/P(B)
P(A∩B) — Independent & Dependent Events
From probabilities or joint table. Marginals, independence test, Venn visualization.
Real-World Scenarios — Click to Load
Input Mode
Venn Diagram (Conceptual)
Probability Bar Chart
Venn Distribution (Doughnut)
Calculation Breakdown
For educational and informational purposes only. Verify with a qualified professional.
📈 Statistical Insights
P(A∩B)
— Joint
P(A∪B)
— Union
P(A|B)
— Conditional
Key Takeaways
- • P(A∩B) = probability both A and B occur — the intersection
- • Independent: P(A∩B) = P(A) × P(B) — events don't influence each other
- • Dependent: P(A∩B) = P(A) × P(B|A) — use conditional probability
- • P(A∪B) = P(A) + P(B) − P(A∩B) — union includes overlap once
- • Joint probability tables: rows × columns, marginals sum to 1
Did You Know?
How It Works
1. Independent Events
P(A∩B) = P(A) × P(B). Dice rolls, coin flips, draws with replacement.
2. Dependent Events
P(A∩B) = P(A) × P(B|A). Cards without replacement, sequential events.
3. Joint Probability Tables
2×2 or 3×3 grid. Each cell = P(row ∩ col). Marginals = row/column sums.
4. Independence Test
If P(A∩B) = P(A)×P(B), events are independent. Check with your computed values.
Expert Tips
Check independence first
If independent, P(A∩B) = P(A)×P(B) — no need for P(B|A).
Use tables for data
When you have counts or proportions, build a joint table.
Venn diagram helps
Overlap = P(A∩B). Union = A + B − overlap.
Marginals must sum to 1
Row sums and column sums each total 1 for valid joint distribution.
Why Use This Calculator vs Other Tools?
| Feature | This Calculator | Conditional | Bayes | Probability |
|---|---|---|---|---|
| P(A∩B) from P(A), P(B) | ✅ | ⚠️ | ⚠️ | ✅ |
| Joint probability table | ✅ | ✅ | ❌ | ❌ |
| Independence test | ✅ | ✅ | ❌ | ✅ |
| Marginal probabilities | ✅ | ✅ | ❌ | ❌ |
| Heatmap & Venn | ✅ | ⚠️ | ❌ | ❌ |
| Educational content | ✅ | ✅ | ✅ | ✅ |
Frequently Asked Questions
What is joint probability?
P(A∩B) = probability that both A and B occur. The intersection of the two events.
When are events independent?
When P(A∩B) = P(A)×P(B). Knowing A occurred doesn't change the probability of B.
How do I compute P(A∩B) for dependent events?
Use P(A∩B) = P(A) × P(B|A). You need the conditional P(B|A).
What is a joint probability table?
A 2×2 or 3×3 grid where each cell is P(row event ∩ column event). Rows and columns sum to marginals.
What are marginal probabilities?
Row and column sums. P(A) = sum of row for A. P(B) = sum of column for B.
How does this relate to conditional probability?
P(A|B) = P(A∩B)/P(B). Joint probability is the numerator.
Can P(A∩B) exceed P(A) or P(B)?
No. P(A∩B) ≤ min(P(A), P(B)) always.
When should I use the table mode?
When you have data in a contingency format or want to explore a full joint distribution.
Joint Probability by the Numbers
Official Data Sources
Disclaimer: This calculator provides joint probability computations for educational and professional reference. For critical applications, verify inputs and consult domain experts.
Related Calculators
Probability Calculator
Calculate single event, multiple event, conditional, and complementary probabilities. Supports union, intersection, and Bayes' theorem calculations with visual probability diagrams.
StatisticsConditional Probability Calculator
Compute P(A|B) = P(A∩B)/P(B) with visual probability tables, tree diagrams, and Venn representations. Three input modes: direct, contingency table, from...
StatisticsProbability of 3 Events Calculator
Calculate combined probabilities for three events using inclusion-exclusion, multiplication rule, and conditional probability.
StatisticsAccuracy Calculator
Compute classification metrics from a confusion matrix: Accuracy, Precision, Recall, F1 Score, Specificity, MCC, and more. Essential for ML model evaluation.
StatisticsBayes' Theorem Calculator
Calculate posterior probabilities using Bayes' theorem. Input prior, likelihood, and evidence to update beliefs with step-by-step Bayesian reasoning.
StatisticsBertrand's Box Paradox
Interactive Bertrand's Box Paradox simulator. Explore why the probability of the other coin being gold is 2/3, not 1/2, with Monte Carlo simulation and Bayesian proof.
Statistics