Probability of 3 Events Calculator
Calculate probabilities for three independent events including union, intersection, and exact occurrences
Three Independent Events Calculator
First independent event
Second independent event
Third independent event
Primary Results
Additional Results
Individual Event Probabilities
Event A
Event B
Event C
Formulas Used
All three occur: P(A ∩ B ∩ C) = P(A) × P(B) × P(C)
At least one occurs: P(A ∪ B ∪ C) = P(A) + P(B) + P(C) - P(A)P(B) - P(A)P(C) - P(B)P(C) + P(A)P(B)P(C)
Exactly one occurs: P(A)P(B')P(C') + P(A')P(B)P(C') + P(A')P(B')P(C)
None occur: P(∅) = P(A') × P(B') × P(C') = (1-P(A))(1-P(B))(1-P(C))
Exactly two occur: P(A)P(B)P(C') + P(A)P(B')P(C) + P(A')P(B)P(C)
Example Calculation
Three Coin Flips Example
Event A: First coin shows heads (50%)
Event B: Second coin shows heads (50%)
Event C: Third coin shows heads (50%)
Question: What are the various outcome probabilities?
Solutions
• All three heads: 0.5 × 0.5 × 0.5 = 12.5%
• At least one head: 1 - (0.5 × 0.5 × 0.5) = 87.5%
• Exactly one head: 3 × (0.5 × 0.5 × 0.5) = 37.5%
• No heads (all tails): 0.5 × 0.5 × 0.5 = 12.5%
Probability Rules for 3 Events
Multiplication Rule
P(A ∩ B ∩ C) = P(A) × P(B) × P(C)
For independent events
Addition Rule
P(A ∪ B ∪ C) includes all overlaps
Subtract intersections, add back triple intersection
Complement Rule
P(A') = 1 - P(A)
Probability of event not occurring
Three Events Tips
Events must be independent for these formulas to work
P(at least one) = 1 - P(none)
Sum of all possible outcomes equals 100%
Use Venn diagrams to visualize overlapping events
Understanding Three Event Probabilities
What Are Independent Events?
Three events A, B, and C are independent if the occurrence of any one (or combination) doesn't affect the probability of the others occurring. This calculator assumes independence for all calculations.
Key Calculations
- •Union (∪): At least one event occurs
- •Intersection (∩): All events occur simultaneously
- •Exclusive: Exactly one event occurs
- •Complement: None of the events occur
Detailed Formulas
All Three Events
P(A ∩ B ∩ C) = P(A) × P(B) × P(C)
At Least One Event
P(A ∪ B ∪ C) = P(A) + P(B) + P(C)
- P(A)P(B) - P(A)P(C) - P(B)P(C)
+ P(A)P(B)P(C)
Exactly One Event
P(A)(1-P(B))(1-P(C))
+ (1-P(A))P(B)(1-P(C))
+ (1-P(A))(1-P(B))P(C)
Real-World Applications
Quality Control
Testing multiple product features simultaneously
Weather Forecasting
Predicting rain, wind, and temperature conditions
Sports Betting
Multiple game outcomes in parlay bets