Four probability tools in one free probability calculator: combine two events (A and B, A or B, conditional, neither), run a binomial distribution, find areas under the normal curve with z-scores, and count permutations and combinations (nPr and nCr). Enter probabilities as decimals, percentages or fractions.
Probability formulas
Worked examples
- Two events: a coin lands heads (½) and a die shows 6 (⅙). They are independent, so P(both) = ½ × ⅙ = 1/12 ≈ 0.0833 and P(at least one) = ½ + ⅙ − 1/12 = 7/12.
- Binomial: the chance of exactly 7 heads in 10 fair flips is C(10,7)·0.5¹⁰ = 120/1024 ≈ 0.117.
- Normal: with IQ scores (μ = 100, σ = 15), about 81.9% of people score between 85 and 130.
- Combinations: choosing 3 people from 10 for a committee gives 10C3 = 120 ways; ranking 3 of them 1st, 2nd and 3rd gives 10P3 = 720.
Permutation or combination?
Ask whether order matters. A PIN code, race podium or seating plan is a permutation. A lottery ticket, pizza toppings or a team pick is a combination. Every combination of r items can be arranged in r! orders, which is why nPr = nCr × r!.
Frequently asked questions
How do I calculate the probability of A and B?
If A and B are independent, multiply: P(A and B) = P(A) × P(B). If they are mutually exclusive, P(A and B) = 0. Otherwise you need the joint probability, which you can enter directly using the “I know P(A and B)” option.
What is the difference between independent and mutually exclusive events?
Independent events do not affect each other (two coin flips). Mutually exclusive events cannot happen together (rolling a 2 and a 5 on one die). Two events with non-zero probabilities can’t be both.
How do I use the binomial probability calculator?
Enter the number of trials n, the probability of success p on each trial and the number of successes k. You get P(X = k) plus the cumulative probabilities P(X ≤ k), P(X ≥ k), P(X less than k) and P(X greater than k), the mean, standard deviation and a chart.
How do I find a probability from a z-score?
Convert x to z = (x − μ)/σ, then find the area under the standard normal curve. Enter the mean, standard deviation and x (and an optional upper bound) to get the left-tail, right-tail and between probabilities.
What is the difference between nPr and nCr?
nPr counts arrangements where order matters; nCr counts selections where it doesn’t. For n = 10 and r = 3, nPr = 720 and nCr = 120.
Related tools
Free educational tool by HawkInc. Results are calculated in your browser and rounded for display; check critical work by hand or with a second method.