Probability Calculator
The Probability Calculator computes event probability, combinations and permutations, and simulates dice or coin rolls. The basic tab calculates P(event), the complement, simplified fraction and odds ratio. The combinations tab computes C(n,r) and P(n,r) for any selection. The dice tab rolls any die type with any count.
How to use the Probability Calculator
Choose a mode and enter values — the result appears immediately.
- Basic probability tabEnter favourable outcomes and total possible outcomes. Furthermore, the probability, its complement, simplified fraction and odds ratio all appear together.
- Combinations and permutations tabEnter n (population size) and r (items to choose). Moreover, both combinations (order irrelevant) and permutations (order matters) are shown side by side.
- Dice simulator tabSelect the die type and number of dice, then click Roll. Additionally, the sum, individual results and min/max range are displayed.
- Interpret odds vs probabilityOdds of 3:1 in favour means 3 wins for every 1 loss — a 75% probability. Furthermore, probability and odds represent the same information in different formats.
- Simplify fractionsThe simplified fraction is shown for the basic probability, making it easier to communicate results verbally — "3 in 8 chance" rather than "37.5%".
Options and variants explained
Probability, odds and fractions are different representations of the same information.
| Representation | Example (rolling a 3 on d6) | Formula |
|---|---|---|
| Probability | 1/6 = 16.67% | Favourable ÷ total |
| Decimal | 0.1667 | Probability as decimal |
| Fraction | 1/6 (simplified) | GCD-reduced numerator/denominator |
| Odds in favour | 1 : 5 | Favourable : unfavourable |
| Complement | 5/6 = 83.33% | 1 − probability |
The formula explained
P(not A) = 1 − P(A) = complement
C(n,r) = combinations: order irrelevant
P(n,r) = n! ÷ (n−r)! = permutations: order matters
The combination formula counts the number of ways to choose r items from n without regard to order. The permutation formula counts the same selection but where different orderings are treated as distinct outcomes. Furthermore, P(n,r) = C(n,r) × r! — permutations equal combinations multiplied by the number of ways to arrange the chosen items.
Worked example: probability of drawing a red card from a standard deck
A standard deck has 52 cards. Red cards: 26. P(red) = 26/52 = 1/2 = 50%. Complement: P(not red) = 50%. Odds in favour: 26:26 = 1:1.
Combinations example: choosing 3 cards from 10 (C(10,3)). C(10,3) = 10! ÷ (3! × 7!) = 3628800 ÷ (6 × 5040) = 120. Furthermore, if order matters (permutations): P(10,3) = 10! ÷ 7! = 720. The 720 permutations collapse to 120 combinations because each combination can be arranged in 3! = 6 ways.
Complementary events and their usefulness
The complement rule (P(not A) = 1 − P(A)) is one of the most useful tools in probability. Instead of calculating a complex probability directly, calculate the probability of the opposite event and subtract from 1. For example, the probability of rolling at least one 6 on two dice is easier found as 1 − P(no sixes) = 1 − (5/6)² = 1 − 25/36 = 11/36.
What is probability?
Probability is the mathematical measure of how likely an event is to occur, expressed as a number between 0 and 1 (or equivalently 0% to 100%). A probability of 0 means the event is impossible; a probability of 1 means it is certain. Furthermore, probability theory provides the foundation for statistics, risk analysis, machine learning and decision-making under uncertainty.
The classical definition of probability — favourable outcomes divided by total equally likely outcomes — applies to simple discrete scenarios like dice, cards and coin flips. Moreover, more advanced probability covers continuous distributions (normal, Poisson, binomial) and conditional probability where events are not independent.
Probability is often expressed as odds in sports betting and in everyday conversation. Odds of 2:1 against mean two unfavourable outcomes for each favourable one — a probability of 1/3. Furthermore, converting between probability and odds is straightforward: P = favourable/(favourable + unfavourable).
Why probability calculations matter
Risk assessment in insurance, finance and medicine relies on probability. An insurer prices a policy based on the probability of a claim. A clinical trial measures whether a treatment effect is likely due to chance or the intervention. Moreover, investment decisions weigh probable returns against probable losses.
Combinations and permutations appear in competitive analysis, scheduling and cryptography. The number of possible passwords of a given length is a permutation problem. The number of possible committees from a group is a combination problem. Furthermore, understanding these counts informs decisions about security and logistics.
Game theory and gambling mathematics are built entirely on probability. Understanding expected value — probability-weighted average outcome — reveals whether a game is fair, advantageous or unfavourable over many repetitions. Moreover, casino games are designed so the house always has a positive expected value.
Common probability mistakes
The gambler's fallacy is treating independent events as though previous outcomes affect future ones. A fair coin does not become more likely to land heads after five tails in a row. Furthermore, each flip is independent — the coin has no memory. Consequently, if you find yourself thinking a streak must end, this fallacy is operating.
Confusing P(A|B) (probability of A given B has occurred) with P(B|A) produces systematic errors. This confusion — the base rate fallacy or prosecutor's fallacy — appears in medical diagnosis interpretation, criminal law and everyday inference. Moreover, Bayes' theorem provides the correct way to update probabilities given new evidence.
Adding probabilities of non-mutually-exclusive events double-counts the overlap. P(A or B) = P(A) + P(B) − P(A and B). Forgetting to subtract the intersection overstates the combined probability. Consequently, always check whether events can occur simultaneously before applying simple addition.
Tips for applying probability correctly
Always define the sample space explicitly — the complete set of equally likely outcomes. Ambiguity about what counts as an outcome is the root of most probability paradoxes. Furthermore, probability paradoxes like the Monty Hall problem are usually resolved by careful enumeration of the sample space.
Simulate when calculation is difficult. Roll the dice simulator many times and observe the distribution of results. Moreover, simulations can verify analytical probability calculations and build intuition for how randomness behaves in practice.
Distinguish between theoretical probability and empirical (experimental) probability. Theoretical: 1/6 chance of rolling a 6. Empirical: observed frequency over many trials. Furthermore, with enough trials, empirical frequencies converge to theoretical probabilities — this is the law of large numbers.
Frequently asked questions
Combinations count selections where order does not matter — choosing 3 team members from 10 gives C(10,3) = 120 options. Permutations count selections where order matters — arranging those 3 in positions gives P(10,3) = 720. Furthermore, permutations always equal combinations × r! for the same n and r.
P(A|B) is the conditional probability of A given that B has already occurred. It is calculated as P(A and B) ÷ P(B). Moreover, conditional probability is fundamental to Bayesian inference, medical diagnostics and spam filtering.
Mutually exclusive events cannot both occur simultaneously. When events can overlap, simple addition double-counts outcomes in the intersection. Furthermore, the inclusion-exclusion principle — P(A or B) = P(A) + P(B) − P(A and B) — provides the correct formula.
Expected value is the probability-weighted average of all possible outcomes. For a fair die: E = (1+2+3+4+5+6)/6 = 3.5. Furthermore, expected value is the foundation of all rational decision-making under uncertainty and determines whether a game is fair.
Each flip has 2 outcomes. For n flips, total outcomes = 2^n. For 10 flips, that is 1,024 outcomes. Furthermore, the probability of exactly k heads follows the binomial distribution: C(n,k) × (1/2)^n.
Related tools
Statistics Calculator
Compute descriptive statistics for a dataset. Furthermore, standard deviation and variance are core to probability.
Percentage Calculator
Convert probabilities to percentages and back. Moreover, express probability in whichever format suits the context.
Quadratic Solver
Solve polynomial equations that arise in probability theory. Additionally, many probability models use quadratic constraints.
Fraction to Decimal
Convert probability fractions to decimals instantly. Furthermore, see probability in all three forms simultaneously.
Scientific Calculator
Perform advanced mathematical calculations for probability. Moreover, factorial and power functions are built in.
Weighted Score Calculator
Apply probability-weighted scoring to decisions. Additionally, expected-value calculations use weighted averages.