Probability Calculator
nCr, nPr, single and two events, binomial, dice and coin probability with steps
Runs entirely in your browser. Nothing you type or upload is sent to us or stored.
Results are for general guidance. Please check anything important before relying on it. Terms of use
How to Use
- Choose what you want to find: combinations (nCr), permutations (nPr), a single event, two events A and B, a binomial probability, the sum of dice, or heads in coin flips.
- Fill in the numbers for that choice, such as n and r, favourable and total outcomes, or P(A) and P(B) as decimals (40% is 0.4).
- The answer updates as you type, as a fraction, decimal and percentage where it makes sense. The formula and the working with your numbers are shown under it, and the dice option adds a table of every possible sum (binomial and coin answers get one too, for up to 60 trials or flips).
- Use copy result or share to send it, or download the report.
A probability calculator with the working shown
This probability calculator covers the questions that come up most in school maths and aptitude tests: counting with nCr and nPr, the probability of a single event, two events that are independent or mutually exclusive, binomial probabilities, dice sums and coin tosses. Each answer comes with the formula and the substituted numbers, so you can match it against your own working for a CBSE board question, a Class 11 permutations exercise or a quantitative aptitude problem in SSC or bank exams.
Combinations and permutations
nPr = n! / (n − r)! order matters
nPr = nCr × r!
Choosing a team of 3 from 10 players is a combination: 10C3 = (10 × 9 × 8)/(3 × 2 × 1) = 120. Picking a captain, vice-captain and wicketkeeper from the same 10 is a permutation, because the roles are different: 10P3 = 10 × 9 × 8 = 720. The calculator works with whole numbers exactly up to n = 170. Big answers are given in scientific notation, with the full digits underneath. The number of 5-card hands from a pack of 52 is 52C5 = 25,98,960.
Single events
When all outcomes are equally likely, P(E) = favourable outcomes ÷ total outcomes, and P(not E) = 1 − P(E). The chance of drawing an ace from a well-shuffled pack is 4/52 = 1/13, about 7.69%. Probability is always between 0 (impossible) and 1 (certain).
Two events: independent or mutually exclusive
| Independent | Mutually exclusive | |
|---|---|---|
| Meaning | One doesn't change the chance of the other | They can't both happen |
| P(A and B) | P(A) × P(B) | 0 |
| P(A or B) | P(A) + P(B) − P(A) × P(B) | P(A) + P(B) |
| Example | Rain today, and a coin landing heads | A die showing 2, and the same die showing 5 |
With P(A) = 0.5 and P(B) = 0.3 for independent events, P(A and B) = 0.15, P(A or B) = 0.65 and P(neither) = 0.5 × 0.7 = 0.35. "At least one" is the same as "A or B", and it is often easiest to find as 1 − P(neither).
Binomial probability
If a trial has two outcomes (success with probability p, failure with 1 − p) and you repeat it n times independently, the chance of exactly k successes is:
mean = np, standard deviation = √(np(1 − p))
Example: a die is thrown 5 times. The chance of exactly two sixes is 5C2 × (1/6)² × (5/6)³ = 10 × 1/36 × 125/216 = 1250/7776 ≈ 0.1608. "At most k" adds the probabilities from 0 up to k, and "at least k" adds them from k up to n.
Dice and coins
With two dice there are 6 × 6 = 36 equally likely outcomes. A sum of 7 can be made in 6 ways, (1, 6), (2, 5), (3, 4), (4, 3), (5, 2) and (6, 1), so P(sum = 7) = 6/36 = 1/6. That makes 7 the most likely total, while 2 and 12 have only 1/36 each. For n fair coin flips, P(exactly k heads) = nCk ÷ 2ⁿ: 5 heads in 10 flips is 252/1024 = 63/256, about 24.6%.
Common mistakes
- Adding probabilities of events that can happen together. Only mutually exclusive events add directly.
- Multiplying probabilities of events that affect each other, like drawing two cards without putting the first back. Those need conditional probability.
- Using nPr when order doesn't matter. Ask yourself whether swapping two chosen items gives a new outcome.