Probability Calculator
Free probability calculator. Calculate the probability of events, conditional probability, union, intersection, and complementary probability with step-by-step explanations.
Enter 0 if events are mutually exclusive
Select a calculation type and enter values to see the solution.
Our free probability calculator helps you calculate probabilities for various scenarios including single events, combined events, conditional probabilities, and Bayes' theorem. Get step-by-step explanations for every calculation.
Understanding Probability
Probability measures how likely an event is to occur, expressed as a number between 0 (impossible) and 1 (certain). It's calculated as:
Probability = Favorable Outcomes ÷ Total Possible Outcomes
Single Event Probability
The most basic probability calculation. For example, the probability of rolling a 4 on a standard die is 1/6 because there's one favorable outcome (rolling a 4) out of six possible outcomes (rolling 1-6).
Probability of A AND B (Intersection)
This calculates the probability that both events occur. For independent events (where one doesn't affect the other):
P(A ∩ B) = P(A) × P(B)
For example, the probability of flipping heads AND rolling a 6 is ½ × ⅙ = ¹⁄₁₂.
Probability of A OR B (Union)
This calculates the probability that at least one of the events occurs:
P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
We subtract the intersection to avoid double-counting outcomes where both occur.
Conditional Probability
The probability of event A occurring given that event B has already occurred:
P(A|B) = P(A ∩ B) ÷ P(B)
For example, the probability of drawing a red card given that it's a heart is 1 (all hearts are red).
Complementary Probability
The probability that an event does NOT occur:
P(A') = 1 − P(A)
If the probability of rain is 0.3, the probability of no rain is 0.7.
Bayes' Theorem
Used to update probabilities based on new information:
P(A|B) = P(B|A) × P(A) ÷ P(B)
Where P(B) = P(B|A) × P(A) + P(B|¬A) × P(¬A)
Bayes' theorem is widely used in medical testing, spam filtering, and machine learning.
Common Probability Examples
| Scenario | Probability | As Percentage |
|---|---|---|
| Flipping heads | ½ | 50% |
| Rolling a 6 | ⅙ | 16.67% |
| Drawing an ace | ⁴⁄₅₂ = ¹⁄₁₃ | 7.69% |
| Two heads in a row | ¼ | 25% |
| At least one 6 in two rolls | ¹¹⁄₃₆ | 30.56% |
Types of Events
- Independent events: One event doesn't affect the other (e.g., consecutive coin flips)
- Dependent events: One event affects the other (e.g., drawing cards without replacement)
- Mutually exclusive: Both cannot occur (e.g., rolling both 1 and 6)
- Complementary: Exactly one must occur (e.g., rain or no rain)