Probability Calculator

This probability calculator works out the likelihood of an event in seconds. Enter your favorable and total outcomes, or plug in two known probabilities, and you get the answer as a decimal, a fraction, and a percentage at the same time. It covers single events, two event problems, three event problems, and situations where the same event repeats across several attempts, so there is no need to work through the addition and multiplication rules by hand every time.

What Is Probability and How Is It Measured?

Probability is a measure of how likely something is to happen. It sits on a scale from 0 to 1, where 0 means the event cannot happen at all and 1 means it is certain. In practice, that same number shows up in a few different outfits depending on where you run into it. A weather app shows it as a percentage, a textbook usually shows it as a fraction, and a calculator like this one often shows it as a decimal by default.

The formula behind all three formats is the same. Take the number of outcomes that count in your favor, divide it by the total number of possible outcomes, and that ratio is your probability.

P(E) = favorable outcomes / total outcomes

Flip a coin and ask for heads, and there is 1 favorable outcome out of 2 total outcomes, which gives 0.5, or 50 percent. That is the entire idea. Every formula further down this page is really just this same ratio applied to a more specific situation.

What Is the Difference Between Theoretical and Experimental Probability?

There are two ways to land on a probability, and mixing them up is a common source of confusion. Theoretical probability is what you would expect based purely on the setup of the problem, worked out before anything actually happens. A fair six-sided die has a theoretical probability of 1 in 6 for rolling a four, and you do not need to touch the die to know that.

Experimental probability comes from actually running the trials and counting the results. Roll that same die 60 times, and you might only land on four 7 or 8 times instead of the expected 10, simply because real outcomes bounce around the expected value, especially over a small number of attempts. Run it a few thousand times instead, and the experimental result tends to settle closer to the theoretical one. This calculator works out theoretical probability directly from the numbers you give it, which is why the result appears instantly rather than needing any trials run first.

How Do You Use the Probability Calculator?

1. Pick the scenario that matches your problem: a single event, two events, three events, or the same event repeated across several trials.

2. Enter your numbers, either as favorable and total outcome counts or directly as a known probability or percentage.

3. If your problem involves two or more events, mark whether they are independent or dependent, since this changes which formula applies.

4. Press "calculate" and read the result as a decimal, a fraction, and a percentage so you can use whichever format fits your work.

How Is the Probability of Two Events Calculated?

Once a second event enters the picture, the question shifts from how likely one thing is to how the two things relate to each other. That relationship decides which formula to use, and picking the wrong one is probably the most common mistake in this whole topic. There are three situations worth knowing here: events that do not affect each other, events where one changes the odds of the other, and cases where you are happy with either one happening.

What Is the Probability of Independent Events A and B?

Two events are independent when the outcome of one has no bearing on the other. Flipping a coin twice is the classic example, since the first flip tells you nothing about the second. For independent events, you multiply the two probabilities together.

P(A and B) = P(A) x P(B)

If P(A) is 0.5 and P(B) is 0.5, then P(A and B) is 0.25, or a 25 percent chance both happen. This calculator applies this rule automatically whenever you mark two events as independent.

What Is the Probability of Dependent Events A and B?

Dependent events change each other. Once the first event happens, it can shrink or shift the pool of outcomes left for the second one, which is exactly what happens when you draw two cards from a deck without putting the first one back. Picture a bag with 10 marbles, 7 black and 3 blue. Draw a blue marble first and remove it, and the odds for the second draw are no longer based on 10 marbles; they are based on 9.

P(A and B) = P(A) x P(B given A)

In the marble example, P(blue first) is 3/10, and P(black second, given blue was already removed) is 7/9, so the combined probability is roughly 0.233, or about 23.3 percent. The most common mistake here is forgetting to shrink the total after the first draw and reusing the original count for both events.

What Is the Probability of A or B (Union)?

Sometimes you only care whether at least one of two events happens, not both. That is the union, written as P(A or B). If the two events can happen together, you add their probabilities and then subtract the overlap so it is not counted twice.

P(A or B) = P(A) + P(B) - P(A and B)

Rolling a die and asking for an even number or a multiple of 3 is a good example. P(even) is 3/6, P(multiple of 3) is 2/6, and P(both) is 1/6, since 6 satisfies both conditions. Add and subtract those and you land on 4/6, or about 66.7 percent. If the two events cannot happen together at all, meaning they are mutually exclusive, the overlap is zero and the formula simply becomes P(A) + P(B).

How Is the Probability of Three or More Events Calculated?

Some problems will not fit neatly into a two-event box. Maybe you are checking three separate machines, tracking three coin flips, or looking at three attempts at a target. The rules do not change much once a third event, C, joins A and B, they just run one step longer.

Take three machines, each with a 20 percent chance of breaking down within a year, all independent of each other. The chance that all three break down is 0.2 x 0.2 x 0.2, which is 0.008, or 0.8 percent. The chance that at least one breaks down works out to close to 49 percent once you run it through the extended addition rule, which is a very different picture from looking at each machine on its own.

How Do You Calculate Probability Across Repeated Trials?

This is a different question from the two and three-event cases above, because here you are not comparing different events; you are repeating the same event several times and asking what happens across those attempts. Three useful results come out of this: the chance the event happens every single time, the chance it never happens, and the chance it happens at least once.

The complement rule does most of the work here. If P is the probability of success on one attempt, then the probability of failure on that attempt is 1 minus P. Raise that failure probability to the power of the number of attempts, and you get the chance of failing every single time. Subtract that from 1, and you get the chance of succeeding at least once.

P(at least one success) = 1 - (1 - P)^n

Roll a die three times and ask for the chance of getting at least one 6. P is 1/6, so 1 minus P is 5/6. Raise that to the third power, and you get about 0.579. Subtract that from 1, and the answer is roughly 42.1 percent, noticeably higher than the 16.7 percent chance on a single roll, which is exactly the kind of intuition this mode of the calculator is built to correct.

How Do Odds Differ From Probability?

Probability and odds get used interchangeably in everyday speech, but they are not the same number. Probability compares favorable outcomes to every possible outcome. Odds compare favorable outcomes to unfavorable outcomes only, ignoring the total.

Converting between the two just needs one small formula. To turn a probability into odds, divide it by 1 minus itself.

Odds = P / (1 - P)

A probability of 0.25 becomes odds of 0.25 / 0.75, which simplifies to 1 to 3. This mix-up is one of the more common reasons people land on a probability page while actually needing an odds conversion, especially anyone coming from sports betting or lottery talk, where the word "odds" gets used loosely to mean "probability."

What Mistakes Cause Wrong Probability Results?

• Mixing up AND with OR, which leads to multiplying when you should add, or the other way around.

• Forgetting to subtract the overlap when two events are not mutually exclusive, which double counts part of the answer.

• Reusing the original total for dependent events instead of shrinking it after the first outcome is removed.

• Entering a percentage where the field expects a decimal, or the reverse, which throws every downstream result off.

• Applying a two-event formula to a repeated-trial problem, which is a different calculation entirely even though it can look similar at a glance.

How Does Probability Connect to Other Statistics Calculators?

Basic probability formulas cover a lot of ground, but a few related problems come up often enough that they deserve their own dedicated tools rather than being crammed into this page. Here is where to go next depending on what you are actually working with.

Converting Odds

If your numbers came in as odds rather than a probability, such as betting odds or lottery odds, you will want a different input format entirely. Our odds calculator handles that conversion both ways, from odds to probability and back.

Repeated Successes in a Fixed Number of Trials

The repeated trials section above only tells you whether an event happens at least once. If you need the odds of an exact number of successes instead, such as guessing correctly on a set number of multiple-choice questions, that calls for a binomial probability calculator instead.

Probability for a Range of Continuous Values

Everything on this page deals with countable, discrete outcomes. When you are instead working with a continuous value, like a height, a test score, or a measurement that can fall anywhere along a range, the shape of the problem changes,B; and you need a normal distribution calculator built around the mean and standard deviation rather than outcome counts.

Counting Outcomes Before You Calculate

A lot of probability problems actually start with a counting question. If you need to know how many ways a group can be selected without caring about order, such as picking 3 cards from a deck of 52, that count is handled by a combination calculator, and the result becomes your total outcomes figure here.

When the order does matter instead, such as ranking the top 3 finishers out of 10 competitors, you need a permutation calculator to get the right outcome count.

Working Backward From Experimental Data

If you are running an actual study or survey and need to know how many responses or trials would make your results trustworthy, that is a separate question from the formulas above. A sample size calculator is the right tool for planning that out before you collect any data.

Frequently Asked Questions

What Is the Formula for Probability?

Probability equals the number of favorable outcomes divided by the total number of possible outcomes, written as P(E) = n(E) / n(S). The result always falls between 0 and 1.

What Are the Basic Rules of Probability?

Every probability sits between 0 and 1. The complement rule says 1 minus P(A) gives the chance A does not happen. For OR, add the two probabilities and subtract any overlap. For AND, multiply the two probabilities together.

What Is the Difference Between Odds and Probability?

Probability compares favorable outcomes to all possible outcomes. Odds compare favorable outcomes to unfavorable outcomes only. A 50 percent probability equals odds of 1 to 1.

How Do You Find the Probability of Two Independent Events Happening Together?

Multiply the two individual probabilities together, since one event does not affect the other. A 0.3 chance and a 0.4 chance combine to a 0.12, or 12 percent, chance of both happening.

Can a Probability Be Greater Than 1 or Negative?

No. A valid probability always falls between 0 and 1. A result outside that range usually points to an input mistake, most often incorrect favorable or total outcome counts.