Multiplying probabilities for “and” events
“And” means multiply along the branches; “or” means add the outcomes.
The common mistake
Adding probabilities when both events must happen — writing P(heads and a 6) as 1/2 + 1/6 — or adding along the branches of a tree diagram.
Instead: Ask whether the question needs both things to happen (and) or either one (or). For “and”, multiply along the path of a tree diagram. For “or” with outcomes that can't both happen, add the finished paths. A probability bigger than 1 means you added where you should have multiplied.
Worked example with Guide me
A coin is flipped and a fair die is rolled. What is P(heads and a 6)?
- Sugar · Does the question need both to happen, or just one?Student · Both — heads and a 6.
- Sugar · So do we add or multiply?Student · Multiply: 1/2 × 1/6.
- Sugar · What does that give?Student · 1/12.
Answer: 1/12.
Try 3 practice questions
1. P(two heads in two coin flips)?
2. A bag has 3 red and 2 blue. Draw one, replace it, draw again. P(red then red)?
3. On a tree, paths A–B have 0.3 and 0.5, and C–D have 0.7 and 0.2. P(A then B or C then D)?
Quick answers
- What is the most common mistake with multiplying probabilities for “and” events?
- Adding probabilities when both events must happen — writing P(heads and a 6) as 1/2 + 1/6 — or adding along the branches of a tree diagram.
- How do I get multiplying probabilities for “and” events right?
- Ask whether the question needs both things to happen (and) or either one (or). For “and”, multiply along the path of a tree diagram. For “or” with outcomes that can't both happen, add the finished paths. A probability bigger than 1 means you added where you should have multiplied.
- Can you show a worked example?
- A coin is flipped and a fair die is rolled. What is P(heads and a 6)? 1/12.
Related concepts
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