How to calculate probability
The probability of an event happening is calculated by dividing the number of favorable outcomes by the total number of possible outcomes, as long as every outcome has the same chance of occurring. The result is always a value between 0 (impossible) and 1 (absolute certainty), which can also be expressed as a percentage by multiplying by 100.
Formulas used
Simple probability: P = favorable outcomes ÷ possible outcomes
"AND" probability (independent events): P(A and B) = P(A) × P(B)
"OR" probability (mutually exclusive): P(A or B) = P(A) + P(B)
"OR" probability (non-exclusive): P(A or B) = P(A) + P(B) − P(A) × P(B)
At least once in n trials: P = 1 − (1 − p)ⁿ
Worked example
The chance of rolling a specific number on a 6-sided die is 1 divided by 6, roughly 16.67%. The chance of rolling at least one "6" across 4 rolls of that same die is 1 minus (5/6) raised to the 4th power, which works out to roughly 51.77%, a classic probability problem known as the Chevalier de Méré's problem.
Independent and mutually exclusive events
Two events are independent when the outcome of one does not affect the chance of the other happening, like the result of two dice rolled together. In that case, the probability of both happening is the product of the individual probabilities. Two events are mutually exclusive when they cannot happen at the same time, like flipping "heads" or "tails" on a single coin toss; in that case, the probability of either happening is simply the sum of the individual probabilities.
Why subtract the overlap in "OR" probability
When two events are not mutually exclusive, meaning they can happen together, simply adding their probabilities would count the overlap between them twice. That's why the general "OR" formula subtracts the probability of both happening together, assuming independence between them. This adjustment makes sure the final result correctly represents the chance of at least one of the two events occurring, without double-counting the case where both occur simultaneously.
Probability across repeated trials
Directly calculating the chance of "at least once" across multiple trials is trickier than it looks, because there are many possible combinations of success. The simplest path is to calculate the opposite: the chance of the event NEVER happening in any of the trials, by raising the failure probability to the power of the number of trials, and then subtracting that value from 1 (100%). That's the method used by the third calculator on this page.