Probability Calculator
Three calculators in one: nCr and nPr, the binomial distribution for a run of trials, and normal distribution probabilities with z-scores.
How it works
Combinations and permutations differ only in whether order matters. Choosing three people for a committee is a combination; choosing a president, secretary and treasurer from the same three is a permutation, and there are always more permutations than combinations for the same n and r.
The binomial distribution covers a fixed number of independent trials each with the same chance of success — coin flips, defect rates, pass rates. The normal distribution covers continuous measurements clustered around a mean, and the z-score simply says how many standard deviations a value sits from that mean.
The formula
Combinations
nCr = n! ÷ (r! × (n − r)!)
Permutations
nPr = n! ÷ (n − r)!
Binomial
P(k) = nCk × pᵏ × (1 − p)ⁿ⁻ᵏ
z-score
z = (x − μ) ÷ σ
Worked examples
| Scenario | Working | Result |
|---|---|---|
| Choose 3 from 10 | 10! ÷ (3!×7!) | 120 combinations, 720 permutations |
| Exactly 4 heads in 10 fair flips | 10C4 × 0.5⁴ × 0.5⁶ | 20.51% |
| IQ of 110, mean 100, SD 15 | z = 0.667 | 74.75% score below it |
When you'd use it
- Working out lottery or card-hand odds
- Checking the chance of a given number of defects in a batch
- Converting a test score to a percentile
- Answering probability coursework questions
Common questions
When do I use combinations rather than permutations?
Use combinations when order does not matter — a committee, a hand of cards, a set of lottery numbers. Use permutations when it does — rankings, podium finishes, assigning distinct roles.
What does a z-score of 2 mean?
The value sits two standard deviations above the mean, which puts it around the 97.7th percentile. Roughly 95% of a normal distribution lies within two standard deviations either side of the mean.
When does the binomial distribution apply?
When there is a fixed number of independent trials, each with only two outcomes and the same probability every time. If trials affect one another — drawing cards without replacement, say — it does not apply.