What is the difference between combinations and permutations?

Combinations count how many ways you can choose items when order does not matter (like picking lottery numbers). Permutations count arrangements where order does matter (like ranking contestants). For the same n and r, permutations are always larger than or equal to combinations because each combination corresponds to multiple orderings.

How do I calculate the probability of two independent events both happening?

Multiply the individual probabilities: P(A and B) = P(A) x P(B). For example, if the probability of rolling a 6 is 1/6 and flipping heads is 1/2, the probability of both happening is 1/6 x 1/2 = 1/12, or about 8.33%.

What is Bayes' theorem used for?

Bayes' theorem updates the probability of a hypothesis based on new evidence. It is widely used in medical testing (calculating the real chance of having a disease given a positive test result), spam filtering, machine learning, and forensic analysis. The formula is P(A|B) = P(B|A) x P(A) / P(B).

What does 'at least one' probability mean?

The probability of at least one success in multiple trials equals 1 minus the probability of zero successes. For example, the chance of rolling at least one 6 in four rolls is 1 - (5/6)^4 = about 51.8%. This complement approach is much simpler than calculating every possible combination of successes.

How accurate is this probability calculator?

This calculator uses exact arithmetic for all computations. Factorials are computed with full precision up to JavaScript's safe integer limit (about 170!). Results are displayed to 10 significant digits. For very large factorials, the calculator uses logarithmic approximation to avoid overflow.

Probability Calculator

Calculate event probability, combinations, permutations, and conditional probability.

Favorable Outcomes

Total Possible Outcomes

Probability

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Multiple Trials

Number of Trials

P(at least one success)

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P(all successes)

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P(no successes)

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Combinations & Permutations

n (total items)

r (items chosen)

Combinations C(n, r)

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Permutations P(n, r)

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Two Independent Events

P(A) — probability of event A

P(B) — probability of event B

P(A and B)

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P(A or B)

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P(not A)

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P(not B)

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How to Use This Probability Calculator

Single event probability: Enter the number of favorable outcomes and total possible outcomes. The calculator instantly shows the probability as a fraction, decimal, and percentage.

Multiple trials: Set the number of trials to see the probability of at least one success, all successes, or zero successes over repeated independent trials.

Combinations and permutations: Enter n (total items) and r (items to choose). C(n,r) gives unordered selections; P(n,r) gives ordered arrangements.

Two events: Enter the probability of each event to calculate joint probability (both happening), union probability (either happening), and complements.

Understanding Probability

Probability measures how likely an event is to occur, expressed as a number between 0 (impossible) and 1 (certain). A probability of 0.5 means the event has an equal chance of happening or not happening. Probability is fundamental to statistics, risk assessment, gambling, insurance, weather forecasting, and countless other fields.

The Basic Probability Formula

The probability of a simple event is: P(E) = favorable outcomes / total possible outcomes. For a fair six-sided die, the probability of rolling a 4 is 1/6 because there is 1 favorable outcome (the 4) out of 6 equally likely outcomes.

Combinations vs Permutations

Both count ways to select items from a larger set, but they differ in whether order matters. Combinations (C(n,r) = n! / (r!(n-r)!)) count unordered selections: choosing 3 cards from a deck of 52 gives C(52,3) = 22,100 possible hands. Permutations (P(n,r) = n! / (n-r)!) count ordered arrangements: the number of ways to arrange 3 books on a shelf from 10 books is P(10,3) = 720.

The Complement Rule

The probability of at least one success in repeated trials is most easily calculated using the complement: P(at least one) = 1 - P(none). For example, the chance of getting at least one head in 3 coin flips is 1 - (0.5)^3 = 0.875 or 87.5%. This is much simpler than calculating P(exactly 1) + P(exactly 2) + P(exactly 3) separately.

Independent vs Dependent Events

Two events are independent if one does not affect the other. Coin flips are independent: the result of the first flip has no effect on the second. For independent events, P(A and B) = P(A) x P(B). Dependent events affect each other: drawing cards without replacement changes the remaining probabilities after each draw.

Frequently Asked Questions

Combinations count selections where order does not matter (choosing a committee). Permutations count arrangements where order does matter (assigning ranks). For the same n and r, permutations are always greater than or equal to combinations.

Use the complement rule: P(at least one) = 1 - P(none). Calculate the probability of zero successes across all trials, then subtract from 1. This is simpler than summing every possible number of successes.

P(A or B) is the probability that at least one of the two events occurs. For independent events, P(A or B) = P(A) + P(B) - P(A) x P(B). The subtraction avoids double-counting the case where both happen.

JavaScript can represent integers exactly up to about 9 x 10^15. Since 170! is approximately 7.3 x 10^306 (near the floating-point maximum), factorials beyond 170 overflow to infinity. The calculator uses this limit to ensure accurate results.

Yes. A typical lottery requires choosing r numbers from n. Use the Combinations section: C(n,r) gives the total number of possible tickets. Your odds of winning are 1/C(n,r). For example, choosing 6 from 49 gives C(49,6) = 13,983,816, so the odds are about 1 in 14 million.