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AQA GCSE · Question 14.1 · Probability
A test consists of two sections, A and B.
- 85% of people pass section A.
- 78% of people who pass section A also pass section B.
- 36% of people who fail section A also fail section B.
Complete the tree diagram.
A test consists of two sections, A and B.
- 85% of people pass section A.
- 78% of people who pass section A also pass section B.
- 36% of people who fail section A also fail section B.
Complete the tree diagram.
How to approach this question
1. Convert the given percentages to decimals.
2. Fill in the known probabilities on the diagram: P(Pass A) = 0.85, P(Pass B | Pass A) = 0.78, P(Fail B | Fail A) = 0.36.
3. Use the rule that probabilities on branches from a single point must add up to 1.
4. Calculate P(Fail A) = 1 - P(Pass A).
5. Calculate P(Fail B | Pass A) = 1 - P(Pass B | Pass A).
6. Calculate P(Pass B | Fail A) = 1 - P(Fail B | Fail A).
7. Write these calculated probabilities on the correct branches.
Full Answer
The completed tree diagram should have the following probabilities on the branches:
- P(Pass A) = 0.85
- P(Fail A) = 1 - 0.85 = 0.15
- P(Pass B | Pass A) = 0.78
- P(Fail B | Pass A) = 1 - 0.78 = 0.22
- P(Fail B | Fail A) = 0.36
- P(Pass B | Fail A) = 1 - 0.36 = 0.64
A tree diagram shows the probabilities of sequential events. The probabilities on any set of branches that start from the same point must sum to 1.
1. **First set of branches (Section A):**
We are given that P(Pass A) = 85% = 0.85.
Therefore, P(Fail A) = 1 - P(Pass A) = 1 - 0.85 = 0.15.
2. **Second set of branches (after passing Section A):**
We are given the conditional probability P(Pass B | Pass A) = 78% = 0.78. (This is the probability of passing B *given that* they passed A).
Therefore, P(Fail B | Pass A) = 1 - P(Pass B | Pass A) = 1 - 0.78 = 0.22.
3. **Third set of branches (after failing Section A):**
We are given P(Fail B | Fail A) = 36% = 0.36.
Therefore, P(Pass B | Fail A) = 1 - P(Fail B | Fail A) = 1 - 0.36 = 0.64.
These are the four values needed to complete the diagram.
Common mistakes
✗ Putting the wrong probabilities on the branches. For example, putting P(Fail A) as 1 - 0.78.
✗ Forgetting to convert percentages to decimals.
✗ Calculation errors, e.g., 1 - 0.85 = 0.25.
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