Conditional Probability Calculator
P(A|B): -
P(B|A): -
P(A∪B): -
Understanding Conditional Probability
What is Conditional Probability?
Conditional probability measures the likelihood of an event occurring given that another event has already occurred.
Key Formulas
- P(A|B) = P(A∩B)/P(B)
- P(B|A) = P(A∩B)/P(A)
- P(A∪B) = P(A) + P(B) - P(A∩B)
- Bayes' Theorem: P(A|B) = P(B|A)P(A)/P(B)
Probability Concepts
Basic Properties
0 ≤ P(A) ≤ 1
P(Sample Space) = 1
P(∅) = 0
Additivity for disjoint events
Independence
P(A|B) = P(A)
P(A∩B) = P(A)P(B)
Mutual independence
Pairwise independence
Chain Rule
P(A∩B∩C) = P(A)P(B|A)P(C|A∩B)
General multiplication rule
Sequential events
Tree diagrams
Total Probability
Law of total probability
Partition of sample space
Marginalization
Expected value calculation
Advanced Applications
Bayesian Statistics
Prior probability
Likelihood function
Posterior probability
Bayesian updating
Machine Learning
Naive Bayes classifier
Hidden Markov models
Probabilistic graphical models
Maximum likelihood estimation
Decision Theory
Expected utility
Risk assessment
Decision trees
Optimal decisions
Real-world Applications
Medical diagnosis
Weather forecasting
Quality control
Financial risk analysis