Probability Calculator
How to use the probability calculator
Choose basic probability, combinations, or permutations. Enter your numbers and get the answer with full working.
Probability formulas
Basic: P = favourable outcomes ÷ total outcomes. Combinations C(n,r) = n! ÷ (r! × (n−r)!), use when order does not matter (lottery numbers). Permutations P(n,r) = n! ÷ (n−r)!, use when order matters (race positions).
Worked examples
Rolling a 6 on a die: P = 1 ÷ 6 ≈ 0.1667 (16.67%). Choosing 2 from 5: C(5,2) = 10 ways. Arranging 2 of 5 in order: P(5,2) = 20 ways.
Frequently asked questions
Ask: does reordering change the outcome? Picking a committee of 3 from 10 is a combination (order irrelevant). Awarding gold, silver, bronze is a permutation (order matters).
There are 10 different ways to choose 2 items from 5 when order does not matter: 5! ÷ (2! × 3!) = 120 ÷ 12 = 10.
No. Probability ranges from 0 (impossible) to 1 (certain). A result above 1 means the inputs were inconsistent.
Complements: the probability that something does not happen
Many questions are easier backwards. Instead of counting the outcomes you want, count the outcomes you do not want and subtract from 1. This is called the complement, and it is the fastest route for any question that contains the words at least one, because the opposite of at least one is none.
The same idea pairs well with counting. Once you know the total number of equally likely outcomes, a probability is just a ratio: favourable outcomes divided by total outcomes. Combinations count the groups when order does not matter, and that count often supplies the numbers for the ratio.
Example: Drawing a marble from a bag of 8
Step 1, Count: A bag holds 5 red marbles and 3 blue marbles, so there are 8 equally likely draws in total.
Step 2, Direct probability: P(blue) = 3/8 = 0.375, which is 37.5%.
Step 3, Complement and counting: P(not blue) = 1 – 3/8 = 5/8 = 0.625. As a counting check, the number of ways to choose any 2 marbles from the 8 is C(8, 2) = 28.
Common mistakes
- Counting ordered outcomes twice: If picking marble A then B is the same result as B then A, use combinations. Switching to permutations inflates the count.
- Letting favourable beat total: Favourable outcomes can never exceed total outcomes. If your fraction is above 1, a count is wrong somewhere.
- Mixing decimals and percents: 0.375 and 37.5% are the same probability written two ways. Move between them by multiplying or dividing by 100, not by 10.
When to use each method
Use basic probability for single draws, rolls, and spins where you can list the outcomes. Use combinations for selections such as committees, hands of cards, and lottery lines, where the order of picking changes nothing. Use permutations for rankings, codes, and finishing orders. And when a direct count looks long, try the complement first: one subtraction often replaces a page of cases.
More questions
Find the probability of none and subtract it from 1. For example, if the chance of no success is 0.3, the chance of at least one success is 1 – 0.3 = 0.7.
Yes. One roll does not change the next, so you multiply the separate probabilities. The chance of rolling a 6 twice in a row is 1/6 * 1/6 = 1/36.
Multiply the decimal probability by 100. So 0.375 becomes 37.5%, and the fraction 3/8 gives the same result once you divide 3 by 8 first.