Probability calculator for normal, binomial and Poisson distributions

This probability calculator works out probabilities from the three distributions you meet most in statistics, normal, binomial and Poisson, and from the chi-square distribution used in hypothesis tests. Choose a distribution, set its parameters, pick the region you want, such as P(X < a) or P(a < X < b), and it shades that region on the graph and gives the probability.

It starts on the standard normal distribution, with mean 0 and standard deviation 1. Each example below lists the exact settings, so you can copy them in and watch the answer appear.

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Probability Calculator

Visualise probability distributions and events

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How to use the probability calculator

1. Under Distribution, choose Normal, Binomial, Poisson or Chi-Sq.
2. Set the parameters: Mean μ and Std deviation σ for a normal distribution, Trials n and Success chance p for a binomial one, Mean rate λ for a Poisson one, or Degrees of freedom for chi-square.
3. In Shade, pick the region: P(X < a), P(X > a), P(a < X < b), or P(X < a or X > b) for both tails.
4. Type Bound a and Bound b, or drag the boundary lines on the graph.
5. Read the answer in the Probability section, written as a statement such as P(X ≤ 3) = 0.2650.

The Formulas section shows the distribution's formula, mean and variance with your numbers in. The Display section switches discrete distributions between bars, lollipops and a line. For practice, Practice and help sets random questions, some asking for a number and some asking you to shade the right region.

Which distribution should you use?

"Less than" or "at most": how the bounds work

Binomial and Poisson variables only take whole numbers, so the wording matters. For these two, the calculator includes the bound: its P(X < a) option gives P(X ≤ a), and its P(X > a) option gives P(X ≥ a). Its answer line always shows which one you have. Turn the words of the question into a bound like this:

For a normal distribution none of this matters: X can take any value, so P(X < a) and P(X ≤ a) are the same.

Example 1: normal distribution probability

A machine fills bags of flour with a mean of 500 g and a standard deviation of 4 g, and the weights are normally distributed. What is the probability that a bag weighs less than 494 g?

Set Normal, μ = 500, σ = 4, shade P(X < a) and a = 494. The calculator gives P(X < 494) = 0.0668, so about 6.7% of bags are under 494 g. By hand you would find the z-score first, how many standard deviations 494 is from the mean:

Then read P(Z < −1.5) = 0.0668 from a table. The calculator shows this z-score with its working under the probability, z = (494 − 500)/4 = −1.50, so you can copy the method. For the share between 495 g and 505 g, shade P(a < X < b) with a = 495 and b = 505: the answer is 0.7887.

Here x is the weight in grams and y is the probability density, as a percentage per gram. The probability is the area under the curve, so P(X < 494) is the small area to the left of 494. Test yourself on a normal distribution question, then press Mark:

Example 2: binomial probability, dice and multiple choice

You roll a fair dice 10 times. What is the probability of at least two sixes? Each roll is a trial with p = 1/6, so X ~ B(10, 1/6). Set Binomial, n = 10, p = 0.16667, shade P(X > a) and a = 2. The calculator gives P(X ≥ 2) = 0.5155.

By hand, "at least two" is quickest with the complement, one minus the chance of zero or one six:

The input takes decimals, so type p to 5 decimal places for thirds and sixths. For exactly two sixes, shade P(a < X < b) with a = 2 and b = 2: the answer is 0.2907.

A second case: a student guesses all 20 questions on a test with 4 choices each, so X ~ B(20, 0.25). The chance of getting 8 or more right is P(X ≥ 8) = 0.1018, about 1 in 10. The expected number right is np = 20 × 0.25 = 5.

Example 3: Poisson probability

A help desk gets an average of 3 calls every 10 minutes, at random. What is the probability of at most 1 call in the next 10 minutes? Set Poisson, λ = 3, shade P(X < a) and a = 1. The calculator gives P(X ≤ 1) = 0.1991.

The chance of a busy spell of 6 or more calls is P(X ≥ 6) = 0.0839. A Poisson model needs the events to happen one at a time, independently, at a steady average rate.

Example 4: a chi-square critical value

A goodness of fit test with 5 degrees of freedom gives a test statistic of 11.07. What is the p-value? Set Chi-Sq, degrees of freedom 5, shade P(X > a) and a = 11.07. The calculator gives P(X > 11.07) = 0.0500, so 11.07 is exactly the 5% critical value for 5 degrees of freedom. A larger statistic gives a smaller p-value and stronger evidence against the null hypothesis.

Chi-square tests always use the right-hand tail, because a big statistic means the observed counts are far from the expected ones. For a test with 3 degrees of freedom, the 5% critical value is 7.815, which the calculator confirms with the same 0.0500.

Mean and standard deviation of each distribution

The Formulas section works these out for the parameters you set. They tell you where a distribution is centred and how spread out it is, which is a quick check that an answer is sensible.

For the test-guessing example, B(20, 0.25) has mean 5 and standard deviation √3.75, about 1.94. Getting 8 or more right is about 1.5 standard deviations above the mean, which fits a probability of around 0.1. For the help desk, Po(3) has mean 3 and standard deviation about 1.73, so 6 or more calls is unusual but not rare.

Basic probability rules the calculator does not do

Many searches for a probability calculator are about two events rather than a distribution. This calculator does not combine events, but the rules are short:

Our guide to how to find probability works through these with examples, and our Probability course teaches each rule with Venn diagrams and tree diagrams. For counting problems, such as the chance of matching 6 numbers chosen from 59 (1 in 45,057,474), use the Permutations and Combinations Calculator. For the long-run average of a game or bet, use the Expected Value Calculator.

Common mistakes

• Mixing up "fewer than" and "at most". For whole-number counts, fewer than 3 means at most 2. Use the table above.
• Using a binomial model when the chance changes. Drawing cards without putting them back changes p each time, so the trials are not independent.
• Typing the variance for σ. The normal distribution needs the standard deviation. If you are given a variance of 16, σ is 4.
• Reading a probability density as a probability. For a normal curve, the height of the curve is not a probability. The area under it is.
• Using the wrong rate for a Poisson question. If calls average 3 per 10 minutes, the rate for an hour is λ = 18. Scale λ to match the time asked about.
• Rounding p too far. p = 0.17 for one sixth gives a noticeably different answer. Keep at least 4 or 5 decimal places.

Practise probability in a short timed paper

Probability distributions come up in AP Statistics practice and in A-level Maths: Statistics in the UK, and basic probability in Geometry, Algebra 2 & SAT Math. Each daily paper is a short timed paper of 8 questions worth 25 to 35 marks, and the papers mix topics.

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