Last updated: May 2026
Calculate probability for single events, combined events, and find combinations or permutations with step-by-step explanations.
| P(A) — Probability of event | – |
| P(A) as percentage | – |
| P(A) as fraction | – |
| P(not A) — Complement | – |
| Odds for (A : not A) | – |
| Odds against (not A : A) | – |
Probability measures how likely an event is to occur, expressed as a number between 0 (impossible) and 1 (certain). It can also be expressed as a percentage (0%–100%) or as odds (e.g., 3:1 against). The fundamental formula is P(event) = favorable outcomes ÷ total possible outcomes. For a standard six-sided die, the probability of rolling a 4 is 1/6 ≈ 16.7% — one favorable outcome out of six equally likely outcomes. The calculator on this page handles single-event, multiple independent events, and binomial (repeated trial) scenarios.
Probability underlies nearly every quantitative decision: insurance pricing, medical diagnostics, weather forecasting, sports betting, and machine learning classifiers all rely on it. Understanding the difference between probability and odds, and between independent and dependent events, prevents the most common reasoning errors — including the gambler's fallacy (believing past outcomes affect future independent ones) and the base rate neglect (ignoring prior probabilities when updating beliefs with new evidence).
| Probability | Percentage | Description | Real-World Example |
|---|---|---|---|
| 0 | 0% | Impossible | Rolling a 7 on a standard die |
| 0.01 | 1% | Very unlikely | Winning a 1-in-100 raffle |
| 0.10 | 10% | Unlikely | Drawing an ace from a shuffled deck (4/52 ≈ 7.7%) |
| 0.25 | 25% | Below average | Rolling a 1 or 2 on a standard die |
| 0.50 | 50% | Even odds | Flipping heads on a fair coin |
| 0.75 | 75% | Likely | Drawing a non-face card from a deck |
| 0.99 | 99% | Near certain | Getting at least one tail in 7 coin flips |
| 1 | 100% | Certain | Drawing any card from a non-empty deck |
What is the difference between probability and odds?
Probability is the ratio of favorable outcomes to total outcomes: P = favorable / total. Odds express the ratio of favorable to unfavorable outcomes: Odds = favorable : unfavorable. A probability of 0.25 (25%) translates to odds of 1:3 (one favorable for every three unfavorable). Gambling markets use odds; statistics and science use probability. To convert: P = odds_for / (odds_for + odds_against).
What is a combination vs. a permutation?
A permutation counts ordered arrangements — the order matters. Choosing 3 people from 10 for President, VP, and Secretary produces 10 × 9 × 8 = 720 permutations. A combination counts unordered selections — the order does not matter. Choosing any 3 people from 10 for a committee produces C(10,3) = 120 combinations. The formula relationship is: C(n,k) = P(n,k) / k! because combinations divide out the k! ways to order the same selection.
What does independent probability mean?
Two events are independent if the outcome of one does not affect the probability of the other. Coin flips are independent — the second flip is unaffected by the first. Drawing cards without replacement creates dependent events — removing a card from the deck changes the probability of each subsequent draw. For independent events: P(A and B) = P(A) × P(B). For dependent events you must adjust the second probability after the first outcome is known.
How is binomial probability calculated?
Binomial probability applies when you have n independent trials, each with the same probability p of success, and you want to know the probability of exactly k successes. The formula is P(X = k) = C(n, k) × p^k × (1−p)^(n−k). C(n,k) counts the number of ways to arrange k successes in n trials; p^k is the probability of those successes occurring; (1−p)^(n−k) is the probability of the remaining trials being failures.
What is the complement rule in probability?
The complement rule states that P(event does NOT occur) = 1 − P(event does occur). It is especially useful when calculating "at least one" probabilities. For example: what is the probability of getting at least one head in 5 coin flips? P(at least one head) = 1 − P(no heads) = 1 − (0.5)^5 = 1 − 0.03125 = 0.96875 or 96.875%. This is much simpler than summing the probabilities of 1, 2, 3, 4, and 5 heads separately.