Probability Calculator
Single, Combined & Combinatorial Probability
Probability = favorable outcomes ÷ total outcomes.
Enter the probability of each independent event (as a decimal from 0 to 1).
Enter a probability (as a decimal from 0 to 1) to find its complement.
Probability of drawing exactly k successes in n draws (without replacement, order doesn't matter) from a population of N items containing K successes.
Understanding Probability
Probability measures how likely an event is to happen, expressed as a number between 0 (impossible) and 1 (certain).
The most basic probability calculation divides the number of outcomes you care about by the total number of equally likely outcomes — like the odds of drawing a specific card from a shuffled deck.
When two events are independent, "and" means multiply, "or" means add and then subtract the overlap. Complements flip a probability to its opposite: what doesn't happen.
The Formulas
Independent AND
Independent OR
Complement
Combinatorial (hypergeometric) probability without replacement:
Why "Without Replacement" Matters
The Combinatorial tab models drawing items from a fixed population without putting them back — like dealing cards from a deck. Each draw changes the composition of what's left, which is why the calculation relies on combinations ("n choose k") rather than simple multiplication of independent probabilities. This is different from repeated independent trials (like coin flips), where the odds never change from one trial to the next.
Every probability falls between 0 and 1, inclusive.
Independent events multiply for "and" probability.
Order-independent selections power combinatorial probability.
Key Takeaways
- Probability is always between 0 and 1. A value outside that range is never valid.
- "And" multiplies, "or" adds and subtracts the overlap — for independent events only.
- Complements are a fast shortcut for "at least one" problems: compute the opposite and subtract from 1.
- Need to compute raw combinations or permutations first? The Permutation and Combination Calculator shows the full factorial breakdown.
Frequently Asked Questions
- Pick the tab that matches your problem: Single Event, Two Events, Complement, or Combinatorial.
- Fill in the fields for that tab.
- Click "Calculate" to see the resulting probability.
For a single event with equally likely outcomes, probability is favorable outcomes ÷ total outcomes. For example, the probability of rolling a 4 on a fair six-sided die is 1 ÷ 6 ≈ 0.1667 (about 16.67%).
For two independent events (the outcome of one doesn't affect the other), P(A and B) = P(A) × P(B) — both must happen. P(A or B) = P(A) + P(B) − P(A) × P(B) — at least one happens; the subtraction avoids double-counting the case where both occur.
The complement of an event is everything that isn't that event. If P(A) is the probability something happens, 1 − P(A) is the probability it doesn't. Complements are useful shortcuts — for example, "at least one success" is often easier to compute as 1 minus "zero successes."
It answers questions like: "If a population of N items contains K successes, what's the probability that a random draw of n items (without replacement, order doesn't matter) contains exactly k successes?" It uses the hypergeometric formula, built from three "n choose k" combinations: C(K, k) × C(N−K, n−k) ÷ C(N, n).
Probability is always a value from 0 (impossible) to 1 (certain), inclusive. Entering something outside that range — like 1.5 or −0.2 — isn't a valid probability, so the calculator rejects it rather than producing a meaningless result.