Probability Calculator
Calculate probability for single events, complements, unions, intersections, and conditional probability. See also Combinations Calculator and Percentage Calculator.
How to Calculate Probability
Probability measures the likelihood of an event occurring, expressed as a number between 0 (impossible) and 1 (certain). For a single event, divide the number of favorable outcomes by the total number of possible outcomes. For combined events, use the addition rule (union), multiplication rule (intersection), or Bayes' theorem (conditional).
Probability Formulas
Single Event: P(A) = Favorable / Total
Complement: P(A') = 1 − P(A)
Union: P(A∪B) = P(A) + P(B) − P(A∩B)
Intersection: P(A∩B) = P(A) × P(B|A)
Independent: P(A∩B) = P(A) × P(B)
Conditional: P(A|B) = P(A∩B) / P(B)
Example Calculation
A bag has 3 red, 4 blue, and 5 green marbles (12 total).
P(red) = 3/12 = 1/4 = 0.25 = 25%
P(not red) = 1 − 0.25 = 0.75 = 75%
Odds for red = 3:9 = 1:3
Odds against red = 9:3 = 3:1
Probability Rules Reference
| Rule | Formula | When to Use |
|---|---|---|
| Addition | P(A∪B) = P(A)+P(B)−P(A∩B) | Either A or B occurs |
| Multiplication | P(A∩B) = P(A)×P(B|A) | Both A and B occur |
| Complement | P(A') = 1 − P(A) | Event does NOT occur |
| Conditional | P(A|B) = P(A∩B)/P(B) | A given B occurred |
Frequently Asked Questions
What is the difference between odds and probability?
Probability is the ratio of favorable outcomes to total outcomes (e.g., 1/4). Odds compare favorable to unfavorable outcomes (e.g., 1:3). They express the same information differently.
What does independent events mean?
Two events are independent if the occurrence of one does not affect the probability of the other. For independent events, P(A∩B) = P(A) × P(B). Example: flipping a coin and rolling a die.
What is conditional probability?
Conditional probability P(A|B) is the probability of A occurring given that B has already occurred. It is calculated as P(A∩B) / P(B).
Can probability be greater than 1?
No. Probability always ranges from 0 (impossible) to 1 (certain). If a calculation gives a value outside this range, check your inputs.
What is the complement rule?
The complement rule states that P(not A) = 1 − P(A). It is useful when calculating the probability of an event NOT happening is easier than calculating it directly.