Probability Basics: Independent Events, Complements, and the 'At Least One' Trick
October 21, 2026
Probability measures how likely an event is, on a scale from 0 (impossible) to 1 (certain), often expressed as a percentage. For equally likely outcomes, the probability of an event is the number of favorable outcomes divided by the total number of outcomes. A fair die has a 1/6 chance of landing on 4. A few simple rules then extend this idea to combinations of events, which covers most everyday probability questions.
The rules are short, but the common mistakes, like assuming events are independent when they are not, or adding where you should multiply, are subtle. This article collects the essentials with worked examples.
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The probability that an event does not happen is 1 minus the probability that it does: P(not A) = 1 − P(A). If the chance of rain is 30%, the chance of no rain is 70%. This rule is the key to many problems, particularly those asking for the chance of 'at least one' occurrence, because it is usually easier to compute the opposite.
Example: the probability of rolling at least one 6 in four rolls of a die. It is easier to find the chance of no 6s in four rolls: (5/6)⁴ = 0.482. The answer is 1 − 0.482 = 0.518, or 51.8%.
Independent Events and the Multiplication Rule
Two events are independent if one does not affect the probability of the other. For independent events, the probability that both happen is the product: P(A and B) = P(A) × P(B). The chance of flipping two heads in a row is 1/2 × 1/2 = 1/4. The chance that all three of three independent machines work, each with 95% reliability, is 0.95³ = 0.857.
Be careful with the assumption of independence. Drawing two cards from a deck without replacement is not independent: the second draw depends on the first. The chance of two aces is 4/52 × 3/51 = 1/221, not (4/52)².
Mutually Exclusive Events and the Addition Rule
Mutually exclusive events cannot happen at the same time. For these, P(A or B) = P(A) + P(B). The probability of rolling a 1 or a 2 on a die is 1/6 + 1/6 = 1/3. When events can overlap, subtract the overlap so you do not count it twice: P(A or B) = P(A) + P(B) − P(A and B). The probability that a card drawn from a standard deck is a heart or a king is 13/52 + 4/52 − 1/52 = 16/52 = 30.8%.
| Situation | Rule | Example |
|---|---|---|
| Not happening | 1 - P(A) | No rain when P(rain) = 0.3: 0.7 |
| Both (independent) | P(A) x P(B) | Two heads: 0.25 |
| Either (exclusive) | P(A) + P(B) | Roll a 1 or 2: 1/3 |
| Either (overlapping) | P(A) + P(B) - P(A and B) | Heart or king: 16/52 |
| At least one | 1 - P(none) | At least one 6 in 4 rolls: 0.518 |
Worked Example: Quality Control
A factory produces items with a 2% defect rate. If you inspect a random sample of 20 items and assume independence, the chance that none is defective is 0.98²⁰ = 66.8%. So the chance of finding at least one defective item is 33.2%. This explains why a small sample can miss a problem with a low defect rate and why inspectors take large samples when defects are costly.
Common Mistakes
Gambler's fallacy: after five heads in a row, the next flip is still 50% heads, because the flips are independent. Confusing P(A and B) with P(A or B). Forgetting that probabilities of mutually exclusive, exhaustive outcomes must sum to 1. And treating dependent events as independent, which can drastically misstate risk. This is how certain financial models underestimated the chance that many loans would default together.
Frequently Asked Questions
What is the probability of at least one success in several trials?
Compute the probability of no successes and subtract it from 1.
When do I multiply probabilities?
For the probability that two or more independent events all occur.
When do I add probabilities?
For the probability that one or another of several mutually exclusive events occurs.
What does independence mean?
That one event's outcome does not change the probability of the other.
Is a coin more likely to land tails after a run of heads?
No. Each flip of a fair coin is independent, so the chance remains 50%.