Find the complement of a probability — the chance that an event does NOT happen.
Since something must either happen or not happen, the two probabilities always add up to exactly 1.
Complement: 1 − P(event)
Every event either happens or it does not. The complement probability calculator takes the probability of an event, P(A), and returns the probability that it does not happen, P(not A) = 1 − P(A). If the chance of rain tomorrow is 0.35, the chance of no rain is 0.65.
The subtraction is simple, but the idea is one of the most useful tools in probability. Many questions ask for at least one success: at least one six in four rolls, at least one defective part in a shipment, or at least one of two backup generators working. These are hard to count directly and easy to solve by finding the chance of none and subtracting from 1. The calculator takes the probability as a decimal between 0 and 1.
1. Write down the probability of the event A as a decimal. Convert a percentage by dividing by 100, and a fraction by dividing numerator by denominator.
2. Apply P(not A) = 1 − P(A).
3. For an at-least-one question, define A as none of the trials succeeds.
4. If the trials are independent, P(none) = P(fail once) multiplied by itself for each trial, that is (1 − p)^n.
5. Then P(at least one) = 1 − (1 − p)^n.
6. Check that P(A) + P(not A) = 1.
The sample space is the set of all possible outcomes, and its total probability is 1. The complement of A contains exactly the outcomes not in A. The two sets do not overlap and together cover everything, so by the addition rule for mutually exclusive events, P(A) + P(not A) = P(sample space) = 1. Rearranging gives P(not A) = 1 − P(A). It holds for any event, whether or not outcomes are equally likely.
Suppose each of 4 independent dice rolls has a 1/6 chance of showing a six. Counting all ways to get one, two, three or four sixes is tedious. Instead, the chance of no six in one roll is 5/6, so in four rolls it is (5/6)^4 ≈ 0.482. The chance of at least one six is therefore 1 − 0.482 ≈ 0.518. This calculator handles the last subtraction; the power step is done separately.
Complementary events are always mutually exclusive, but mutually exclusive events are not always complements. Getting a 1 or a 2 on a die cannot happen together, yet together they cover only a third of the outcomes. Independence is a different idea again: it means one event does not change the chance of another. An event and its complement are never independent unless one of them is certain, because knowing A happened tells you for sure that not A did not.
A delivery app says 88% of its orders in Rohit's area arrive on time, so he enters 0.88 to find the chance that tonight's order is late.
Answer: Probability the event does NOT happen 0.12
Entering a percentage such as 88 instead of the decimal 0.88.
Treating two events as complements when they do not cover all outcomes, such as win and lose in a match that can be drawn.
Computing at least one as n × p instead of 1 − (1 − p)^n, which can give answers above 1.
Using the complement of at least one as none of them fail, instead of none of them succeed.
Forgetting that the at-least-one shortcut with powers assumes the trials are independent.
Finding the chance of at least one success in repeated trials, such as exam attempts or sales calls.
Reliability checks, such as the chance that at least one of several backup systems works.
Turning a known success rate into a failure rate for planning and risk assessment.
Weather and sports predictions, where the chance of the opposite result is needed.
Solving the birthday problem, via the chance that no two people share a birthday.
Why is the complement useful?
Sometimes it is easier to find the chance of something NOT happening, and subtract that from 1, instead of working out the event directly. This is common in 'at least one' probability problems.