Probability Calculator
Calculate probability for coin tosses, dice rolls, card draws and normal distributions. Get combinations, permutations and z-scores.
Exact probability
24.6094%
Cumulative (≤ k)
62.3047%
Last updated: January 2026
How this calculator is verified
Checked by Rahim Virani on
- Complement events verified to sum to exactly 1
- Independent-events multiplication used; conditional probability flagged as a different calculation
- Fractional inputs outside 0-1 rejected with the valid range stated
The full verification method is on our how we verify page. Found an error? Tell us and we will re-check it.
When to Use This Calculator
Probability problems decide board exam marks, placement test results and, increasingly, everyday decisions, and this calculator handles the standard toolkit: permutations, combinations, binomial probability and the normal distribution. A student checking homework on drawing balls from a bag, an aspirant practising probability for CAT or bank exams, and a data analyst sanity-checking a binomial calculation before writing code all get exact answers with the formula shown. It computes P(X = k) and cumulative probabilities for a given n and p, plus z-scores and normal CDF values for statistics questions. Enter the parameters and the tool shows both the numeric answer and the method, so it teaches while it calculates.
How to Use This Calculator
- Step 1: Pick the scenario type: coin tosses, dice rolls, card draws or a normal distribution.
- Step 2: Enter the parameters, such as the number of tosses and the number of heads you want, or the mean and standard deviation for the normal case.
- Step 3: Read the exact probability and the cumulative probability of your outcome.
- Step 4: For card scenarios, review the combinations and permutations shown, which count the possible ways the draw can happen.
Worked Example
For a fair coin tossed 10 times, the probability of exactly 5 heads is 252 divided by 1024, about 24.6%, computed with the binomial formula. The tool also handles dice rolls, card draws and the normal distribution: for example, the number of possible 5-card hands from a 52-card deck is 2,598,960. Choose the scenario, enter the parameters, and the result includes the exact probability, the cumulative probability, and for card scenarios the combinations and permutations as well.
Tips and Common Mistakes
- •Tip 1: Use the cumulative probability when you want the chance of 'at most' or 'at least' a certain count, such as 3 or fewer heads.
- •Tip 2: For card problems, remember that combinations count unordered hands, so 5 cards from 52 gives 2,598,960 possible hands.
- •Tip 3: Treat real-world events cautiously: probability models assume a fair coin and a fair die unless you adjust the inputs.
- ✗Mistake 1: Expecting 5 heads in 10 tosses to be a 50% event, when the exact probability is only about 24.6%.
- ✗Mistake 2: Using permutations for a card hand problem when order does not matter, which overcounts the possibilities enormously.
Frequently Asked Questions
What is probability?
Probability is the likelihood of an event occurring, expressed as a number between 0 and 1. A probability of 0.5 means a 50% chance.
What is the difference between combinations and permutations?
Combinations (nCr) count groups where order doesn't matter. Permutations (nPr) count arrangements where order matters.
What is conditional probability?
It is the probability of one event happening given that another has already occurred, written P(A|B). The tool's Bayes mode uses it: P(A|B) = P(B|A)·P(A) ÷ P(B), which is how doctors interpret test results.
How do I find the probability of two events both happening?
For independent events, multiply the individual probabilities: P(A and B) = P(A) × P(B). For a coin flipped twice, the chance of heads both times is 0.5 × 0.5 = 0.25, or 25%.
What is the difference between permutations and combinations?
Permutations count ordered arrangements — choosing 2 from 3 letters gives AB, BA, AC, CA, BC, CB (6 ways). Combinations ignore order — the same choice gives AB, AC, BC (3 ways).
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