Conditional Probability Calculator
Calculate conditional probability ($P(A|B) = \frac{P(A \cap B)}{P(B)}$), joint probability, and test event independence.
Calculate Conditional Probability P(A|B)
Enter joint probability P(A ∩ B) and condition probability P(B).
Calculation Results
Calculated using conditional probability definitions: P(A|B) = \frac{P(A \cap B)}{P(B)}
Quick Summary
The Conditional Probability Calculator evaluates conditional probability $P(A|B) = \frac{P(A \cap B)}{P(B)}$ and checks whether events A and B are independent.
Formula Explanation
P(A|B) = \frac{P(A \cap B)}{P(B)} \quad \text{where } P(B) > 0
\text{Independence Criterion: } P(A|B) = P(A)How It Works
Conditional probability calculates the updated probability of event A after knowing event B has occurred. The Conditional Probability Calculator divides joint probability $P(A \cap B)$ by condition probability $P(B)$, verifying statistical independence.
Step-by-Step Worked Example
Practical Problem: In a student group, 30% take Spanish ($P(B) = 0.30$) and 15% take both Spanish and French ($P(A \cap B) = 0.15$). Calculate probability a student takes French given they take Spanish ($P(A|B)$).
- Step 1: Identify joint probability $P(A \cap B)$: $P(A \cap B) = \mathbf{0.15}$.
- Step 2: Identify condition probability $P(B)$: $P(B) = \mathbf{0.30}$.
- Step 3: Divide joint probability by condition probability: $P(A|B) = 0.15 / 0.30 = \mathbf{0.5000\text{ (50.00\%)}}.$
- Step 4: Test for Independence: If unconditional $P(A) = 0.40$, $P(A|B) = 0.50 \neq 0.40$.
- Step 5: Interpretation: Taking Spanish increases French enrollment probability from 40% to 50% (dependent events).
Real-World Calculation Examples
Scenario 1: Language Enrollment (P(A∩B)=0.15, P(B)=0.30)
Parameters: P(A∩B) = 0.15, P(B) = 0.30
Result: P(A|B) = 0.5000 (50.00% conditional probability).
Scenario 2: Medical Diagnostic Test Accuracy
Parameters: True positive P(T∩D) = 0.045, Disease P(D) = 0.05
Result: P(T|D) = 0.9000 (90% Sensitivity).
Scenario 3: E-Commerce Purchase Given Cart Add
Parameters: P(Purchase∩Cart) = 0.20, P(Cart) = 0.50
Result: P(Purchase|Cart) = 0.4000.
Scenario 4: Card Deck Probability (Ace given Red)
Parameters: P(Red Ace) = 2/52, P(Red Card) = 26/52
Result: P(Ace|Red) = 2/26 = 1/13.
Key Benefits of Using This Calculator
Instant Conditional Updating
Updates event probabilities based on known condition evidence.
Statistical Independence Testing
Compares $P(A|B)$ with $P(A)$ to verify independent vs dependent events.
Joint & Marginal Probability Sync
Integrates joint $P(A \cap B)$ and marginal $P(B)$ inputs.
100% Free & Client-Side
Executes locally in your browser with zero latency or web server transmission.
Frequently Asked Questions (FAQ)
What is conditional probability $P(A|B)$?
Conditional probability $P(A|B)$ is the probability that event A occurs given that event B has already occurred.
What is Multiplication Rule of Probability?
$P(A \cap B) = P(B) \cdot P(A|B) = P(A) \cdot P(B|A)$.
How do I test if two events A and B are independent?
Events A and B are independent if $P(A|B) = P(A)$ or $P(A \cap B) = P(A) \cdot P(B)$.
What is difference between $P(A|B)$ and $P(B|A)$?
$P(A|B)$ conditions on B; $P(B|A)$ conditions on A. In general, $P(A|B) \neq P(B|A)$ (Confusion of the Inverse fallacy!).
Can conditional probability be calculated if $P(B) = 0$?
No — division by zero is undefined; conditional probability requires $P(B) > 0$.
How does Bayes Theorem relate to conditional probability?
Bayes Theorem reverses conditional probability: $P(A|B) = \frac{P(B|A) P(A)}{P(B)}$.
What is Law of Total Probability?
$P(B) = \sum P(B|A_i) P(A_i)$ across a partition of sample space.
How do I calculate conditional probability in Excel?
Use division formula =P_A_and_B / P_B.
What is Monty Hall Problem?
The Monty Hall problem is a famous conditional probability puzzle where switching doors doubles your win probability from 1/3 to 2/3.
What is Simpson's Paradox?
Simpson's paradox occurs when a conditional probability trend in several groups reverses when groups are combined.