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Probability calculator

Five calculators in one: the chance of an event (and over many tries), two events with and / or / given, exact combinations and permutations, the binomial distribution and the normal distribution — each with a chart and the full working.

  • Free
  • No sign-up
  • Private — runs in your browser

How do I calculate a probability?

  1. 1Pick what you want to work out: one event, two events, combinations, binomial or normal.
  2. 2Type your numbers — probabilities can be decimals (0.25), percentages (25%) or fractions (1/4).
  3. 3Read the answer as a decimal and a percentage, with the related probabilities underneath.
  4. 4Check the chart and the step-by-step working, which uses your own numbers.

What are the main probability formulas?

  1. 01P(A) = favourable ÷ total · P(not A) = 1 − P(A)
  2. 02P(A or B) = P(A) + P(B) − P(A and B) · P(A | B) = P(A and B) ÷ P(B)
  3. 03nCr = n! ÷ (r!(n − r)!) · nPr = n! ÷ (n − r)!
  4. 04Binomial: P(X = k) = C(n, k) · pᵏ · (1 − p)ⁿ⁻ᵏ, mean np, SD √(np(1 − p))
  5. 05Normal: z = (x − μ) ÷ σ, P(X < x) = Φ(z)

A probability is a number from 0 (impossible) to 1 (certain). For equally likely outcomes it is the number of outcomes you want divided by the number possible: one six out of six faces is 1/6 ≈ 0.1667. The chance of something happening at least once in several independent tries is 1 minus the chance it never happens: at least one six in four rolls is 1 − (5/6)⁴ ≈ 0.518.

For two events, 'and' means both happen and 'or' means at least one does. If A and B are independent, P(A and B) = P(A) × P(B); if they are mutually exclusive, P(A and B) = 0. The addition rule subtracts the overlap so it isn't counted twice. Conditional probability P(A | B) is the chance of A once you know B has happened.

Combinations count choices where order doesn't matter, permutations where it does. The answers grow very fast — 52C5 is 2,598,960 poker hands — so this calculator works them out exactly with whole-number arithmetic, digit for digit, instead of rounding.

The binomial distribution counts successes in n independent tries with the same chance p each time; the normal distribution is the bell curve. Probabilities here are computed to about 15 significant digits (much more precise than a printed z-table), and everything runs in your browser.

Good to know

  • 'At least one' problems are easiest backwards: 1 − P(none).
  • Use combinations when the order of the chosen items doesn't matter (a hand of cards), permutations when it does (a race result).
  • Only multiply probabilities for 'and' when the events are independent; otherwise use P(A) × P(B | A).
  • For a normal-table question, set μ = 0 and σ = 1 and type the z-score as the value.

Probability questions

How do you calculate probability?

For equally likely outcomes: probability = number of favourable outcomes ÷ total number of outcomes. The chance of drawing a heart from a full deck is 13 ÷ 52 = 0.25, or 25%.

What is the difference between nCr and nPr?

nCr counts combinations, where order doesn't matter: choosing 3 of 10 people gives 120. nPr counts permutations, where order matters: 1st, 2nd and 3rd place among 10 runners gives 720. nPr = nCr × r!.

How many 5-card poker hands are there?

52C5 = 52! ÷ (5! × 47!) = 2,598,960.

How do you find P(A or B)?

Add the two probabilities and subtract the chance that both happen: P(A or B) = P(A) + P(B) − P(A and B). For mutually exclusive events the overlap is 0, so you just add.

When do you use the binomial distribution?

When there is a fixed number of independent trials, each with only two outcomes (success or failure) and the same chance of success every time — e.g. the number of heads in 10 coin tosses.

How do I find the probability from a z-score?

Choose the normal distribution with mean 0 and standard deviation 1, then type the z-score as the value: P(Z < 1.96) = 0.975. For any normal distribution, the calculator first works out z = (x − μ) ÷ σ.

Sources

Formulas last reviewed: September 30, 2026 ·How we check our formulas

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