Conditional Probability Calculator
Calculate conditional probability P(A|B) using Bayes' theorem and probability rules
Calculate Conditional Probability
Enter value between 0 and 1 (e.g., 0.05 for 5%)
Probability of B occurring when A has occurred
Probability of B occurring when A has NOT occurred
Calculation Results
Complete Probability Table
Formulas used:
Medical Testing Example
Disease Prevalence Example
Disease prevalence in population: 5% (P(D) = 0.05)
Test sensitivity (true positive rate): 91% (P(+|D) = 0.91)
Test specificity (true negative rate): 95% (P(-|D̄) = 0.95)
False positive rate: 5% (P(+|D̄) = 0.05)
Question
If a random person tests positive, what's the probability they actually have the disease?
Solution
P(D|+) = P(D∩+) / P(+)
P(+) = 0.05 × 0.91 + 0.95 × 0.05 = 0.093
P(D∩+) = 0.05 × 0.91 = 0.0455
P(D|+) = 0.0455 / 0.093 ≈ 48.92%
Probability Tree Concept
Key Concepts
Conditional Probability
P(A|B) = probability of A given B has occurred
Bayes' Theorem
Relates P(A|B) to P(B|A)
Law of Total Probability
P(B) = Σ P(B|Ai) × P(Ai)
Applications
Medical diagnosis, quality control, machine learning
Understanding Conditional Probability
What is Conditional Probability?
Conditional probability quantifies the likelihood of an event occurring given that another event has already occurred. It's denoted as P(A|B), read as "probability of A given B."
Why is it Important?
- •Essential for medical diagnosis and screening tests
- •Used in machine learning and artificial intelligence
- •Critical for risk assessment and quality control
- •Helps update beliefs with new information
Mathematical Foundation
Real-World Applications
- Medical Testing: Disease probability given test results
- Spam Filtering: Email classification based on content
- Weather Forecasting: Rain probability given cloud conditions
- Quality Control: Defect probability given inspection results