---
title: "Probabilities Sum to One"
description: "Understand that when all possible outcomes of a trial are listed, their probabilities must add up to 1; use this to find the probability of an event NOT happening: P(not A) = 1 − P(A); apply this shor"
canonical: https://lightmysky.com/learn/mathematics/probabilities-sum-to-one-mt_3fwYu7imd4
source: https://lightmysky.com/learn/mathematics/probabilities-sum-to-one-mt_3fwYu7imd4.md
retrieved: 2026-10-19
---

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# Probabilities Sum to One

Understand that when all possible outcomes of a trial are listed, their probabilities must add up to 1; use this to find the probability of an event NOT happening: P(not A) = 1 − P(A); apply this shortcut to avoid counting all unfavourable outcomes directly

Subject: Mathematics · Area: Probability · Ages 10 to 11
Page: https://lightmysky.com/learn/mathematics/probabilities-sum-to-one-mt_3fwYu7imd4

## Ready when they can

- List all outcomes of spinning a 4-colour spinner and verify their probabilities add up to 1
- Calculate P(not rolling a 3) as 1 − 1/6 = 5/6 using the complement rule
- Spot an error in a probability table where the values don't sum to 1 and explain what's wrong

## Lesson: All the chances add up to one

The trials in this lesson have a finite list of possible outcomes, like the colours on a spinner. Finite means the list ends. Exactly one outcome happens in each trial. If the list includes every possible outcome, their probabilities add to 1. Together they make a whole. To find one missing chance, add the known chances and subtract their total from 1.

*(drawing: Red, blue, green, yellow: 0.5 + 0.2 + 0.2 + 0.1 = 1.)*

P(A) means the probability of event A. Sometimes you want the chance that A does not happen, like not landing on red. Use P(not A) = 1 − P(A). Take the chance of A away from 1, and you get the chance of everything else.

**Example.** A spinner has 8 equal sections and 1 of them is gold, so P(gold) = 1/8. To find P(not gold), subtract from 1: 1 − 1/8 = 7/8. Seven of the eight sections are not gold.

**Tip.** You never need to count every 'not' outcome by hand. Subtract the event's probability from 1.

**Recap.** All outcomes together always add to 1, so P(not A) = 1 − P(A).

## Practice

20 questions on this page, each with its working shown.

## Needs first

- [The 0-to-1 Probability Scale](https://lightmysky.com/learn/mathematics/the-0-to-1-probability-scale-mt_t0g2SlP404)
- [Calculating Simple Probability](https://lightmysky.com/learn/mathematics/calculating-simple-probability-mt_Zt30Gxi-qp)

## Opens up

- [The Addition Rule and Mutually Exclusive Events](https://lightmysky.com/learn/mathematics/the-addition-rule-and-mutually-exclusive-events-mt_QsIusFefeE)
- [Complementary events](https://lightmysky.com/learn/mathematics/complementary-events-mt_XfyqXLqzpx)