Probability Calculator: Events, Odds & Combinations
Calculate basic event probability, combined AND/OR probability for two events, and combinations or permutations (nCr / nPr), all instantly in your browser.
How this probability calculator handles single events, combined events, and counting problems
Probability questions generally fall into three families: how likely is one thing to happen, how do two probabilities combine, and how many ways can you arrange or choose items from a group. This tool treats each as its own mode with its own formula, rather than forcing every question through one generic calculation, since the correct math genuinely differs between them.
Everything computes client side. The combinations and permutations mode uses JavaScript’s BigInt type for factorials, so results stay exact rather than drifting into floating point rounding once numbers get large.
Mode 1: single-event probability
Mode 2: combined events, AND and OR
For two events A and B, the tool applies different rules depending on whether you mark them independent or mutually exclusive. Independent events, like two separate coin flips, use multiplication for AND and inclusion exclusion for OR. Mutually exclusive events, ones that literally cannot both happen, like rolling a 2 and a 5 on the same single die roll, skip the AND calculation entirely, since it is always zero.
The subtraction in the independent-OR formula matters more than it looks. Simply adding P(A) and P(B) double counts the overlap where both events occur simultaneously. Subtracting P(A and B) removes that double count exactly once, which is the inclusion exclusion principle in its simplest two set form.
Mode 3: combinations and permutations
| Formula | Meaning | Order matters? |
|---|---|---|
| nPr = n! / (n−r)! | Permutations | Yes |
| nCr = n! / (r! × (n−r)!) | Combinations | No |
Simplified fraction display
Single-event results are shown as a reduced fraction alongside the percentage, using the same GCD-based simplification found across this site’s other math tools, so 50/200 displays as 1/4.
Sanity check on mutually exclusive inputs
If you mark two events mutually exclusive but their probabilities sum past 1, the tool flags it, since two genuinely mutually exclusive events can never have probabilities that add past certainty.
- Probability axioms covers the foundational rules, including the complement and inclusion-exclusion principles used above.
- Combination and permutation cover the counting formulas behind mode 3.
Questions probability answers
Working out odds in a card or dice game, estimating the chance of two independent risks both occurring in a project plan, calculating lottery or raffle odds using combinations, figuring out how many possible arrangements exist for a schedule or seating chart, and checking basic statistics homework involving compound probability. Anywhere uncertainty needs to be quantified as a number between 0 and 1, or counted as a number of possible arrangements, one of these three modes covers it.
FAQ: Probability Calculator
Divide the number of favorable outcomes by the total number of possible outcomes: P(event) = favorable ÷ total. For example, the probability of rolling a 4 on a standard six-sided die is 1 favorable outcome divided by 6 total outcomes, or about 16.67%.
Independent events don’t affect each other’s probability, like flipping a coin twice, the first flip’s result has no bearing on the second. Mutually exclusive events cannot both happen at the same time, like a single coin flip landing on both heads and tails simultaneously. These are opposite concepts and use different formulas, so mixing them up is a very common probability mistake.
By definition, mutually exclusive events cannot occur together, so the probability of both happening simultaneously is always zero. This is different from independent events, where P(A and B) equals P(A) × P(B), which is generally greater than zero unless one of the individual probabilities is zero.
A combination counts selections where order doesn’t matter (choosing 3 toppings for a pizza, regardless of the order you name them), while a permutation counts arrangements where order does matter (assigning gold, silver, and bronze medals to 3 of 10 racers). Permutations are always greater than or equal to combinations for the same n and r, since every combination can typically be arranged in multiple orders.
Odds for an event are expressed as the ratio of favorable outcomes to unfavorable outcomes (not favorable to total, which is probability), so odds of 1:5 mean 1 favorable outcome for every 5 unfavorable ones. Odds against simply flips that ratio. Odds and probability describe the same underlying likelihood but are conventionally expressed differently, especially in betting contexts.
No, valid probabilities are always between 0 and 1 inclusive (or 0% to 100%), a probability of 0 means the event is impossible and 1 means it’s certain. If a calculation produces a value outside that range, it usually indicates invalid inputs, like probabilities for mutually exclusive events that don’t actually sum to something sensible.
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