Probability · Ages 14–15
Tree diagrams
Multiply along a path; without replacement, one fewer the second time.
How to do it
Method
Multiply along a path; without replacement, one fewer the second time
3R 2B, no replacement: 35 × 24 = 310
Another way
Counting ordered outcomes
(3 × 2) ÷ (5 × 4) = 620 = 310
In full
When something happens in steps, list every outcome on a tree diagram: each step's branches are that step's outcomes, and a path from left to right to the end is one outcome. Two coins (H heads, T tails) give HH, HT, TH, TT, 2 × 2 = 4 outcomes; exactly one head is HT or TH, so 24 = 12. With balls, be clear: with replacement, the second draw's probabilities are the first's; without, the bag has one ball fewer. A bag of 3 red (R) and 2 blue (B), two drawn without replacement: P(both red) = 35 × 24 = 620 = 310, multiplying the probabilities along the path. Another way: treat every ball as different and count ordered outcomes: 5 × 4 = 20 in all, 3 × 2 = 6 with both red, so P = 620 = 310, the same.
Worked examples
Have a go first, then open each one for the answer and the working.
Example 1
A family has 3 children, each as likely a boy (B) as a girl (G), as the tree diagram shows. The tree has outcomes; of them have exactly 2 girls.
Fill in each blank
Show the answer and working
Answer: 8, 3
Each step splits in 2, so 3 steps make 8 outcomes; those with exactly 2 G number 3.
Example 2
A bag holds 4 red balls (R) and 4 blue (B). One is drawn and not put back, then another, as the tree shows. The probability that both are red is .
Answer in simplest form
Show the answer and working
Answer: 314
Without replacement: red first 48, then 3 red of 7 left: 48 × 37 = 314.
Example 3
Tossing 5 coins, the probability that all land the same way (all heads or all tails) is 116.
True or false?
Show the answer and working
Answer: True
The tree has 25 = 32 outcomes, only 2 all the same: 232 = 116.
Practice questions to try
Try these 5; the answers are at the end. For a whole worksheet, see VIP worksheets below.
1.A family has 3 children, each as likely a boy (B) as a girl (G), as the tree diagram shows. The tree has outcomes; of them have exactly 0 girls.
Fill in each blank
Answer:
2.There are 3 roads from A to B and 3 from B to C. Going from A to C through B, there are different routes; choosing one at random, the probability of Ka-yee's favourite is .
Fill in each blank
Answer:
3.A bag holds 6 red balls (R) and 1 blue (B). One is drawn and not put back, then another, as the tree shows. The probability that both are red is .
Answer in simplest form
Answer:
4.A bag holds 2 red balls (R) and 6 blue (B). One is drawn, its colour noted and put back, and another drawn, as the tree shows. The probability that both are red is .
Answer in simplest form
Answer:
5.A bag holds 6 red balls (R) and 2 blue (B). One is drawn and not put back, then another, as the tree shows. The probability of one red and one blue, in either order, is .
Answer in simplest form
Answer:
All the answers
- 1. 8, 1
- 2. 9, 19
- 3. 57
- 4. 116
- 5. 37
VIP practice notes and papers
Aimed at tests and exams: questions on this topic, new every time, to print or save as a PDF.
- Practice notes: The key point, worked examples step by step, then questions of the same kind to follow, hints fading
- Practice paper: Set out in parts like a school paper, marked, with answers and full working
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