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Free Online Probability Calculator, Two Events and Repeated Trials

Work out the probability of two events happening together or separately, the chance of at least one success across repeated tries, and conditional probability. Whether the events are independent is asked rather than assumed, because assuming it quietly is where most wrong answers in probability come from.

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Free Online Probability Calculator, Two Events and Repeated Trials

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🎯 Probability Calculator Two events, or many tries · independence asked, not assumed

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What are you working out

Reading —

🎯 Two events

A probability between 0 and 1, or a percentage with a per cent sign. Both are understood.

Chance of A

Chance of B

Does one affect the other?

Try one of these

The answer

—

Both—

Either, or both—

Neither—

Exactly one—

As odds—

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How to Work Out a Probability

Three steps, and the first one is the one that matters.

Decide whether the events affect each other Two coin flips do not. Two cards drawn without putting the first back do. This is the question that decides whether multiplying is allowed, and it is asked first for that reason.

Type the chances, as decimals or percentages Either 0.25 or 25% works. If the events do affect each other you will also be asked for the chance of the second one once the first has happened.

Read all five answers, not just the one you came for Both, either, neither, exactly one, and the odds. People usually arrive wanting one of these and leave realising they wanted a different one.

What to Know Before You Multiply

Independence, overlap, and the mistake nearly everyone makes.

Multiplying two probabilities only works if they are independent. The chance of two things both happening is the chance of the first times the chance of the second given the first. When the first does not affect the second, that second number is just the plain chance and the familiar multiplication appears. Drawing two hearts from a deck without replacing the first is a quarter times twelve fifty firsts, not a quarter times a quarter, and the difference is small enough that nobody notices it is wrong.

For "either" you have to subtract the overlap. The chance of A or B is the chance of A plus the chance of B minus the chance of both, because adding counts the cases where both happen twice over. Leaving out that last term is the most common slip in probability, and when both chances are large it produces an answer above one, which is impossible and at least visible. When they are small it produces an answer that is merely wrong.

At least one in several tries is not the chance times the number of tries. A three in ten chance tried four times is not twelve in ten. The way to get it is to work out the chance of failing every time, which is seven tenths to the fourth, and subtract from one. That gives about 76 per cent. Multiplying gives 120 per cent, and the fact that it is obviously impossible is the only thing that saves you.

Percentages and decimals are both accepted, and the per cent sign is respected. Type 25% or 0.25 and you get the same thing. Type 25 on its own and it is read as 25, which is not a probability, and you are told so rather than having it silently divided by a hundred. Guessing what someone meant is how a tool ends up giving a confidently wrong answer.

Odds and probability are different things. A probability of a quarter is odds of one to three, not one to four. Probability compares the successes to all the outcomes; odds compare the successes to the failures. Betting and medical statistics use odds, most other fields use probability, and the two get swapped constantly, so both are shown.

This does not check whether your numbers are consistent with each other. If you say the chance of B given A is higher than is possible for the chance of B you have typed, the arithmetic still runs, because there are real situations with that shape and this page cannot know which you are in. The note under the answer points out when the numbers imply something unusual, but the judgement is yours.

Key Features & Capabilities

What it does, and what it deliberately does not.

Independence asked, not assumed The first question is whether one event affects the other, because that is what decides whether multiplying is valid.

Five answers at once Both, either, neither, exactly one, and the odds, because people often arrive wanting a different one than they thought.

At least one in n tries Worked out the right way round, by subtracting the chance of failing every time, with the wrong way shown for comparison.

Percentages understood Type 25% or 0.25. A bare 25 is reported as out of range rather than quietly treated as 25 per cent.

The working shown Each formula with your own numbers in it, including the term people leave out of the "either" calculation.

Nothing is uploaded It runs in this tab and keeps working offline.

About Probability

Probability is unusual among school topics in that the arithmetic is trivial and the mistakes are universal. Almost nobody gets the multiplication wrong. Almost everybody multiplies when they should not, adds without subtracting the overlap, or works out the chance of at least one success by multiplying the chance by the number of tries. All three produce answers that look reasonable.

The independence question is the important one. The chance of two things both happening is the chance of the first times the chance of the second given that the first happened. When the first has no effect on the second, that second number is just its own plain probability, and you get the familiar multiplication. When it does have an effect, you do not. Drawing two hearts from a deck without putting the first back is a quarter times twelve fifty firsts, which is about 5.9 per cent rather than the 6.25 per cent that multiplying gives. The error is small, which is exactly why it survives.

The second is the overlap. The chance of A or B happening is not the sum of the two, because the sum counts the outcomes where both happen twice. Subtracting the chance of both fixes it. If the two chances are large enough, forgetting this produces an answer above one hundred per cent and you notice. If they are small it produces an answer that is slightly too big and you do not.

The third is the one that comes up most often in real life. A three in ten chance, tried four times: what is the chance of at least one success? Not four times three tenths, which is more than certainty. You work out the chance of failing every single time, seven tenths to the fourth power, and take it away from one, which gives about 76 per cent. Both answers are shown here side by side, because seeing the impossible one next to the correct one is the fastest way to remember which is which.

Frequently Asked Questions

Independence, "at least one", odds and percentages.

Only when the events are independent, meaning the first happening does not change the chance of the second. Coin flips and dice rolls are independent. Cards drawn without replacement, or anything where one outcome uses something up, are not.

Because adding the two chances counts the outcomes where both happen twice over. Subtracting the chance of both removes the duplication. Forgetting it is the most common error in probability and can give an answer above one hundred per cent.

One minus the chance of failing every time. For a chance p over n tries that is one minus (1 minus p) to the power of n. It is not p times n, which can exceed one and therefore cannot be a probability.

Yes, with the per cent sign: 25% is read as 0.25. A bare 25 is treated as the number 25, which is not a valid probability, and you are told so rather than having it quietly divided by a hundred.

Probability compares successes to all outcomes, odds compare successes to failures. A probability of a quarter is odds of one to three, not one to four. Both are shown because the two are swapped so often.

The chance of one event given that another has already happened, written P(B given A). It is what you need when the events affect each other, and it is the number that changes when a card is not replaced or an item is used up.

No. Everything is worked out in your own browser and the page keeps working with the network off.