Probability Calculator

What Is Probability?

Probability measures how likely an event is, as a number from 0 to 1. A probability of 0 means the event cannot happen, 1 means it is certain, and 0.5 means it is as likely as not. It is often shown as a percentage, so 0.25 is a 25% chance. In the simplest case, probability is the number of desired outcomes divided by the total number of equally likely outcomes. A fair die shows a 4 with probability 1/6. This probability calculator works out how two events combine, and how a series of independent events behaves.

Probability of Two Events

With two independent events A and B — meaning one does not influence the other — these rules apply:

• Both occur (intersection): P(A ∩ B) = P(A) × P(B)
• At least one occurs (union): P(A ∪ B) = P(A) + P(B) − P(A) × P(B)
• Neither occurs: P(not A and not B) = (1 − P(A)) × (1 − P(B))
• Exactly one occurs: P(A) + P(B) − 2 × P(A) × P(B)

If P(A) = 0.5 and P(B) = 0.25, the chance both happen is 0.125, and the chance at least one happens is 0.625. Enter probabilities between 0 and 1.

Worked Example

Two independent events each with a 50% probability (like two coin flips both landing heads): the probability both happen is 25%, and the probability at least one happens is 75%.

A Series of Independent Events

The chance of an event happening every time in n independent trials is pⁿ, and the chance it happens at least once is 1 − (1 − p)ⁿ. Flipping a coin three times, the chance of three heads is 0.5³ = 12.5%, but the chance of at least one head is 1 − 0.5³ = 87.5%. This second form explains why a small risk repeated many times becomes likely.

Normal Distribution

Many real-world measurements, such as height or test scores, follow a bell-shaped normal distribution. The probability of falling within a certain range equals the area under the curve over that range, which depends on the mean and standard deviation. Roughly 68% of values lie within one standard deviation of the mean and 95% within two. For related calculations see the Z-Score Calculator and the Standard Deviation Calculator.

Common Mistakes

• The gambler's fallacy: past outcomes of independent events do not change the next one — a coin that came up heads five times is still 50/50.
• Adding probabilities of events that overlap without subtracting the overlap.
• Assuming events are independent when they actually influence each other.

To count outcomes before computing probabilities, use the Permutation and Combination Calculator.