Probability Calculator
Calculate probability, complementary probability, union and intersection of events, conditional probability, combinations, permutations, and more. Perfect for students, statisticians, data analysts, and anyone working with uncertainty.
What Is Probability?
Probability is a measure of the likelihood that a specific event will occur. It is expressed as a number between 0 (impossible) and 1 (certain). For example, the probability of flipping a fair coin and getting heads is 0.5 or 50%.
The basic formula for probability is:
P(A) = Number of favorable outcomes Ć· Total number of possible outcomes
For example: The probability of rolling a 4 on a standard 6-sided die is 1 Ć· 6 ā 0.1667 or 16.67%.
This calculator supports several probability concepts:
- Basic Probability: Simple probability of a single event.
- Complementary Probability: Probability that an event does NOT occur.
- Union & Intersection: Probability of A or B occurring (union) or both A and B occurring (intersection).
- Conditional Probability: Probability of A given that B has occurred.
- Combinations & Permutations: Number of ways to select or arrange items.
- Dice Simulator: Roll virtual dice and see the results.
Key Probability Formulas
- Complement: P(A') = 1 ā P(A)
- Union: P(A āŖ B) = P(A) + P(B) ā P(A ā© B)
- Intersection: P(A ā© B) = P(A) Ć P(B) if independent
- Conditional: P(A|B) = P(A ā© B) Ć· P(B)
- Combinations: C(n,r) = n! Ć· (r! Ć (nār)!)
- Permutations: P(n,r) = n! Ć· (nār)!
ā Probability Calculator FAQ
What is probability?
Probability is a measure of the likelihood that an event will occur. It ranges from 0 (impossible) to 1 (certain). It can also be expressed as a percentage or odds.
How do I calculate basic probability?
Use the formula: P(A) = Favorable Outcomes Ć· Total Outcomes. For example, the probability of drawing a heart from a standard deck is 13 Ć· 52 = 0.25 or 25%.
What is complementary probability?
The complement of an event A is the probability that A does NOT occur: P(A') = 1 ā P(A). For example, if P(A) = 0.3, then P(A') = 0.7.
What is the difference between union and intersection?
Union (A āŖ B) is the probability that at least one of the events occurs. Intersection (A ā© B) is the probability that both events occur.
What is conditional probability?
Conditional probability P(A|B) is the probability of event A occurring given that event B has already occurred. Formula: P(A|B) = P(A ā© B) Ć· P(B).
What is the difference between combinations and permutations?
Combinations (C(n,r)) count the number of ways to select r items from n items without regard to order. Permutations (P(n,r)) count the number of ways to select and arrange r items from n items where order matters.
What is the formula for combinations?
C(n,r) = n! Ć· (r! Ć (nār)!). For example, C(5,2) = 5! Ć· (2! Ć 3!) = 10.
What is the formula for permutations?
P(n,r) = n! Ć· (nār)!. For example, P(5,2) = 5! Ć· 3! = 20.
What are the laws of probability?
The two main laws are: 1) The sum of probabilities of all possible outcomes is 1. 2) The probability of an event is between 0 and 1 inclusive.
What is the difference between mutually exclusive and independent events?
Mutually exclusive events cannot occur at the same time (e.g., rolling a 1 or a 6 on a die). Independent events have no influence on each other (e.g., coin flips).
What is the expected value in probability?
Expected value (EV) is the average outcome of a random event over many trials. It is calculated as the sum of each outcome multiplied by its probability.
What is the probability of rolling a sum of 7 with two dice?
There are 6 ways to roll a sum of 7 out of 36 total outcomes. Probability = 6/36 = 1/6 ā 16.67%. The dice simulator can show this.
What is the probability of getting exactly 3 heads in 5 coin flips?
This is a binomial probability problem: P(X=3) = C(5,3) Ć 0.5Ā³ Ć 0.5Ā² = 10 Ć 0.125 Ć 0.25 = 0.3125 or 31.25%.
What is Bayes' theorem?
Bayes' theorem describes the probability of an event based on prior knowledge of conditions that might be related to the event. Formula: P(A|B) = P(B|A) Ć P(A) Ć· P(B).
What is the difference between probability and odds?
Probability is the ratio of favorable outcomes to total outcomes. Odds are the ratio of favorable outcomes to unfavorable outcomes. For example, probability 0.25 = odds 1:3.
How do I convert probability to odds?
Odds = P Ć· (1 ā P). For example, P = 0.25 gives odds = 0.25 Ć· 0.75 = 1:3.
What is the central limit theorem?
The central limit theorem states that the distribution of sample means approaches a normal distribution as sample size increases, regardless of the shape of the population distribution.
Is this probability calculator free?
Yes, this calculator is completely free to use. No registration or personal data storage is required. All calculations are performed in your browser.