Probability of Dependent Events Calculator
Calculate probability of dependent events ($P(A \cap B) = P(A) \cdot P(B|A)$), sampling without replacement, and dependent event chains.
Calculate Dependent Event Probabilities
Enter probability of 1st Event P(A) and conditional probability P(B|A).
Calculation Results
Calculated using dependent event multiplication rules: P(A \cap B) = P(A) \cdot P(B|A)
Quick Summary
The Probability of Dependent Events Calculator evaluates joint probability $P(A \cap B) = P(A) \cdot P(B|A)$ for sequential dependent trials.
Formula Explanation
P(A \cap B) = P(A) \cdot P(B|A)
P(A \cap B \cap C) = P(A) \cdot P(B|A) \cdot P(C|A \cap B)How It Works
Dependent events occur sequentially where the outcome of event A alters the conditions for event B. The Probability of Dependent Events Calculator multiplies prior probability $P(A)$ by conditional probability $P(B|A)$.
Step-by-Step Worked Example
Practical Problem: Drawing 2 aces consecutively from a deck of 52 cards without replacement. Calculate joint probability $P(A \cap B)$.
- Step 1: Calculate 1st card ace probability $P(A)$: $P(A) = 4 / 52 = \mathbf{0.076923}$.
- Step 2: Update deck state for 2nd draw: 51 cards remaining, 3 aces remaining.
- Step 3: Calculate 2nd card ace probability $P(B|A)$: $P(B|A) = 3 / 51 = \mathbf{0.058824}$.
- Step 4: Multiply probabilities ($P(A \cap B)$): $P(A \cap B) = (4/52) \times (3/51) = 12 / 2,652 = \mathbf{0.004525\text{ (0.45\%)}}.$
- Step 5: Interpretation: The probability of drawing 2 consecutive aces without replacement is 0.45% (1 in 221).
Real-World Calculation Examples
Scenario 1: Two Aces Drawn Without Replacement
Parameters: P(A) = 4/52, P(B|A) = 3/51
Result: P(A∩B) = 0.0045 (0.45% chance).
Scenario 2: Job Interview & Hiring Sequence
Parameters: Pass interview P(A) = 0.40, Hired given interview P(B|A) = 0.70
Result: P(Hired) = 0.2800 (28.00% overall chance).
Scenario 3: Marbles drawn from Bag Without Replacement
Parameters: 5 red / 10 total (P(A)=5/10), 4 red / 9 total (P(B|A)=4/9)
Result: P(Both Red) = 20/90 = 0.2222.
Scenario 4: Sequential Quality Testing Failure
Parameters: P(Stage 1 fail) = 0.10, P(Stage 2 fail|Stage 1) = 0.50
Result: P(Both fail) = 0.0500.
Key Benefits of Using This Calculator
Sequential Dependent Chain Modeling
Models probability chains for multi-step sampling without replacement.
Dependence Effect Ratio
Computes dependence ratio $\frac{P(B|A)}{P(A)}$ to quantify directional impact.
Tree Diagram Integration
Calculates branch probabilities for probability decision trees.
100% Free & Client-Side
Executes locally in your browser with zero latency or web server transmission.
Frequently Asked Questions (FAQ)
What are dependent events?
Dependent events are events where the outcome of the first event changes the probability of the second event.
What is multiplication rule for dependent events?
$P(A \cap B) = P(A) \times P(B|A)$.
Why does sampling without replacement create dependent events?
Removing an item reduces total sample size $N$ and favorable count $K$, altering probabilities for subsequent draws.
How do I calculate dependent event probabilities in Excel?
Use multiplication formula =P_A * P_B_given_A.
What is probability tree diagram?
A probability tree diagram visually maps sequential conditional probability branches ($P(A)$, $P(B|A)$, $P(C|A \cap B)$).
How do 3 dependent events multiply?
$P(A \cap B \cap C) = P(A) \cdot P(B|A) \cdot P(C|A \cap B)$.
What is difference between $P(A \cap B)$ and $P(B|A)$?
$P(A \cap B)$ is the probability both happen out of ALL outcomes; $P(B|A)$ is the probability B happens out of ONLY A outcomes.
Can dependent events become independent if population N is large?
Yes — when population $N$ is very large (e.g. 100,000), sampling without replacement closely approximates independent sampling with replacement.
What is conditional dependence?
Conditional dependence occurs when two events become dependent only when conditioned on a third variable.
What is Markov chain property?
A Markov chain assumes the next state depends ONLY on the current state, independent of prior historical path.