About Probability Calculator
This probability calculator is free, needs no sign-up and runs in your browser, so nothing you enter is uploaded. Give it the probabilities of two events A and B and say whether they are independent, mutually exclusive or have a known overlap. It returns P(A and B), P(A or B), P(not A), P(not B), the two conditional probabilities and the chance of exactly one or neither, with a Venn diagram.
Switch to Conditional for P(A | B) from P(A and B) and P(B), to Bayes' theorem to update a prior probability after evidence such as a test result, or to Repeated trials for questions like "what is the chance of at least one six in four rolls" and "exactly three heads in ten flips".
You can type probabilities as decimals (0.25), percentages (25%) or fractions (1/4). Every answer comes with the formula written out with your numbers, so it doubles as a worked solution.
How to use Probability Calculator
- Choose a calculation: Two events, Conditional, Bayes' theorem or Repeated trials.
- Enter the probabilities as decimals, percentages or fractions, such as 0.4, 40% or 2/5.
- For two events, say how A and B are related: independent, mutually exclusive, or P(A and B) known.
- Read the headline answer and the table of related probabilities. The Venn diagram shows each region.
- Open the working to see each formula with your numbers filled in.
Features
- P(A and B), P(A or B), complements, conditional probabilities and exactly one or neither.
- Independent, mutually exclusive and known-overlap cases, with a check that the numbers are possible.
- Conditional probability with an independence check.
- Bayes' theorem with the total probability P(B) and a count table out of 10,000 people.
- Repeated trials: none, at least one, exactly k, at most k and at least k, using the binomial distribution.
- Venn diagram in SVG that scales to your screen.
- Decimals, percentages and fractions accepted everywhere.
The main probability rules
The addition rule counts the overlap once. The multiplication rule for independent events multiplies the chances. Conditional probability restricts to the cases where B happened, and Bayes' theorem turns P(B | A) around into P(A | B).
P(A or B) = P(A) + P(B) - P(A and B)
independent: P(A and B) = P(A) x P(B)
P(A | B) = P(A and B) / P(B)
P(not A) = 1 - P(A)
Bayes: P(A | B) = P(B | A) P(A) / P(B)
P(B) = P(B | A) P(A) + P(B | not A) P(not A)
at least one in n trials = 1 - (1 - p)^nWorked examples
Independent events, P(A) = 0.5 and P(B) = 0.4 gives P(A and B) = 0.2, P(A or B) = 0.7
0.5 x 0.4 = 0.2, and 0.5 + 0.4 - 0.2 = 0.7. Neither has probability 0.3.
P(A) = 0.6, P(B) = 0.5, P(A and B) = 0.4 gives P(A | B) = 0.8, P(B | A) = 0.666667
0.4 / 0.5 = 0.8 and 0.4 / 0.6 = 0.666667. They are not independent, since 0.6 x 0.5 = 0.3, not 0.4.
Bayes: 1% have a condition, the test finds 90% of them and wrongly flags 9% of the rest gives P(condition | positive) = 0.091743
P(positive) = 0.9 x 0.01 + 0.09 x 0.99 = 0.0981. Only about 9.2% of positives are true, because the condition is rare.
At least one six in 4 rolls of a die gives 0.517747
1 - (5/6)^4 = 1 - 0.482253.
Exactly 3 heads in 10 flips gives 0.117187
C(10, 3) x 0.5^10 = 120 / 1024.
Good to know
Independent is not the same as mutually exclusive
Mutually exclusive events cannot both happen, so P(A and B) = 0. Independent events do not affect each other, so P(A and B) = P(A) x P(B). So when both events have a positive probability, mutually exclusive events can never be independent. Knowing that one mutually exclusive event happened tells you the other did not, which is as dependent as it gets.
Why a positive test is often still unlikely to be right
Bayes' theorem accounts for how common the thing is before the test. When a condition is rare, even a fairly accurate test produces many more false positives than true ones. The natural frequency table in Bayes mode makes this visible: out of 10,000 people, count the true positives and the false positives side by side.
Privacy: how it works
- All calculations run in your browser tab. The probabilities you enter are not sent to Ulkaa or to any other server.
Tips
- Probabilities must be between 0 and 1. The calculator says which input is out of range.
- For "at least one", it is easier to find the chance of none and subtract from 1.
- If the overlap you enter is impossible for the given P(A) and P(B), the tool shows the allowed range.
- Repeated trials assume each trial is independent with the same chance.
Frequently asked questions
Is it really free?
Yes. There is no price, no trial and no credit card. Ulkaa Tool does the work with your own device, so there is no server bill to pass on to you.
Do I need an account?
No. There is no sign-up and no login. Open the page and use it. An Ulkaa account is not needed for anything on this tool.
Does it run in my browser?
Yes. Your probabilities are processed by code running in your browser tab, on your own device.
Is my data uploaded?
No. Your probabilities are never sent to Ulkaa or to anyone else. You can check this yourself by watching the Network tab in your browser developer tools while you use the tool.
How do I calculate the probability of A and B?
If the events are independent, multiply the probabilities. If not, use P(A and B) = P(A) x P(B | A).
How do I calculate P(A or B)?
Add the two probabilities and subtract the overlap: P(A) + P(B) - P(A and B).
What is conditional probability?
It is the chance of A once you know B happened. It is P(A and B) divided by P(B).
What is Bayes' theorem used for?
It updates a probability after new evidence. It converts P(B | A), such as a test sensitivity, into P(A | B), the chance the condition is present given a positive result.
How do I find the probability of at least one success?
Calculate the chance of none, (1 - p) to the power n, and subtract it from 1.
Can I enter fractions or percentages?
Yes. Type 1/4, 25% or 0.25. All three mean the same thing.
How are exactly k successes calculated?
With the binomial formula: C(n, k) x p^k x (1 - p)^(n - k).
Popular Probability Calculator pages
- Conditional Probability CalculatorEnter P(A), P(B) and P(A and B) to get P(A given B) and P(B given A).
- Bayes' Theorem CalculatorEnter a prior, a sensitivity and a false positive rate to get the updated probability.
- At Least One Probability CalculatorEnter the chance of one trial and the number of trials to get the chance of at least one success.