Probability: independent and dependent events; tree diagrams
Tree diagrams appear on Papers 2 and 3 of OCR J560. Higher-tier extends to conditional probability and dependent events (without replacement). These are reliable marks if you know the multiplication and addition rules.
Basic probability rules
- P(A happens) + P(A does not happen) = 1.
- All probabilities on a complete set of branches sum to 1.
Independent events
Two events are independent if the outcome of one does not affect the probability of the other.
Multiplication rule for independent events: P(A and B) = PA × PB.
Example: P(head) = 0.5; P(six) = 1/6. P(head AND six) = 0.5 × 1/6 = 1/12.
Addition rule (mutually exclusive events)
P(A or B) = PA + PB — only when A and B are mutually exclusive (cannot both happen).
Example: P(six) = 1/6; P(five) = 1/6. P(five or six) = 1/6 + 1/6 = 1/3.
Tree diagrams
Use a tree diagram when there are two or more sequential events.
Rules:
- Multiply along branches to find P(this path).
- Add probabilities of paths that satisfy the condition.
Example: A bag contains 3 red and 2 blue balls. A ball is drawn, replaced, then another is drawn.
| Red (3/5) | Blue (2/5) | |
|---|---|---|
| Red (3/5) | RR: 9/25 | RB: 6/25 |
| Blue (2/5) | BR: 6/25 | BB: 4/25 |
P(both same colour) = P(RR) + PBB = 9/25 + 4/25 = 13/25.
Dependent events (without replacement)
When items are NOT replaced, probabilities change on the second draw.
Example: Bag has 3 red, 2 blue. Draw two without replacement.
First draw red (3/5) → second draw red is now (2/4) because one red is gone. P(both red) = 3/5 × 2/4 = 6/20 = 3/10.
Second draw blue given first was red: P = 2/4 = 1/2. P(red then blue) = 3/5 × 2/4 = 6/20 = 3/10. P(blue then red) = 2/5 × 3/4 = 6/20 = 3/10. P(different colours) = 3/10 + 3/10 = 3/5.
Common OCR exam mistakes
- Forgetting to update probabilities on second draw in without-replacement problems.
- Adding when should multiply (and vice versa): multiply along branches (AND); add paths (OR).
- Branch probabilities not summing to 1 — always check.
- Using fractions and decimals inconsistently — pick one and stick to it.
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