Addition Rule of Probability

1Summary 重點摘要

Learn how to calculate 'or' probability — chance of either event happening.

2Vocabulary cards Vocabulary

3Step by step Steps

1. Step 1: Mutually exclusive events and the addition ruleTwo events that cannot happen in the same trial are mutually exclusive: on one die, rolling a 1 and rolling a 2 cannot occur together. For such events the probabilities simply add: P(A or B) = P(A) + P(B). Always confirm exclusivity before using the formula — it is the first judgement step of this topic, and the formula fails without it.
2. Step 2: A mutually exclusive exampleRoll a fair die and find the probability of getting 1 or 2. Each face has probability 16, so P(1) = 16 and P(2) = 16; the events are mutually exclusive, giving P(1 or 2) = 16 + 16 = 26 = 13. As a check, only two of the six outcomes (1 and 2) qualify, and 26 = 13 agrees with the formula. Similar questions, such as 'a 5 or a 6', are handled the same way once exclusivity is confirmed.
3. Step 3: Correcting for non-mutually-exclusive eventsWhen two events can happen together they are not mutually exclusive: 'an even number' and 'a 6' share the outcome 6, since 6 is itself even. Adding directly would count the overlap twice, so the intersection must be subtracted once: P(A or B) = P(A) + P(B) − P(A and B). The test is a single question — can both events happen at the same time? If yes, use the corrected formula.
4. Step 4: A non-mutually-exclusive exampleDraw one card from a deck of 52 and find the probability of a heart or an Ace. P(heart) = 1352 and P(Ace) = 452; the Ace of hearts belongs to both events, so P(heart and Ace) = 152. The events are not mutually exclusive, hence P(heart or Ace) = 1352 + 452 − 152 = 1652 = 413. Forgetting to subtract 152 gives the wrong answer 1752 — the classic trap of this question type.
5. Step 5: How to judge exclusivityJudge in three steps: write down both event definitions precisely; search for an outcome satisfying both at once, such as the Ace of hearts being both a heart and an Ace; if such an outcome exists the events are not mutually exclusive and the intersection is subtracted, and only when none exists may you add directly — never judge by feeling. Exam questions with 'at least', such as 'passes at least one subject', mean 'Maths or English', which calls for the addition rule rather than adding the two probabilities blindly.
6. Step 6: Venn diagrams and exam formatDraw two partially overlapping circles for A and B: separate circles mean mutually exclusive, and the overlap is exactly P(A and B). Fill each region with its probability, writing the overlap only once; adding all occupied regions then gives P(A or B) without double counting. For DSE working: state whether the events are mutually exclusive and why, quote the formula, substitute the numbers, and finish with a simplified fraction or a percentage.

4Diagrams Diagrams

5Worked example Example

6Common mistakes Mistakes

7Quick check Quick check

8Recap Recap

• Mutually exclusive: P(A or B) = P(A) + P(B)
• Not mutually exclusive: P(A or B) = P(A) + P(B) - P(A and B)
• First check if mutually exclusive, then use formula
• At least one = A or B
• Venn diagram helps understand overlap