1. GCSE: Tree Diagrams and Frequency Trees
Skipton Girls’ High School

2. !Mutually Exclusive Events
If A and B are mutually exclusive events, they can’t happen at the same time. Then:
P(A or B) = P(A) + P(B)
! Independent Events
If A and B are independent events, then the outcome of one doesn’t affect the other. Then:
P(A and B) = P(A) × P(B)

3. Tree Diagrams Unconditional
Question: There are 3 blues and 5 reds on a spinner – There are 2 spins.
Spin 2
P(bb)  3/8 x 3/8 = 9/64
Spin 1
P(blue) = 3/8
P(blue) = 3/8
P(br)  3/8 x 5/8 = 15/64
P(red) = 5/8
P(rb)  5/8 x 3/8 = 15/64
P(blue) = 3/8
P(red) = 5/8
P(red) = 5/8
P(rr)  5/8 x 5/8 = 25/64
Check: = 64

4. R R B R B B Tree Diagrams Conditional 4 ? 6 5 ? 7 2 ? 6 5 ? 6 2 ? 7 1
Question: Given there’s 5 red balls and 2 blue balls. What’s the probability that after two picks we have a red ball and a blue ball?
Key Point: Note that probabilities go on the lines, and events at the end.
After first pick, there’s less balls to choose from, so probabilities change. 4 6 ? R ? 5 7 R B ? 2 6 5 6 R ? 2 7 ? B 1 6 B ?

5. Tree Diagrams
Question: Give there’s 5 red balls and 2 blue balls. What’s the probability that after two picks we have a red ball and a blue ball?
We multiply across the matching branches, then add these values. 4 6 R 5 7 R 5 21 ? B 2 6 5 21 5 6 R ? 2 7 B 1 6 B 10 21 ?
P(red and blue) =

6. Conditional and Unconditional Tree Diagrams
...with replacement:
The item is returned before another is chosen. The probability of each event on each trial is fixed.
...without replacement:
The item is not returned.
Total balls decreases by 1 each time.
Number of items of this type decreases by 1.
Note that if the question doesn’t specify which, e.g. “You pick two balls from a bag”, then PRESUME WITHOUT REPLACEMENT.

7. Example (on your sheet)? 3 8 ? 3 8 ? 5 8 ?
5 8 × 5 8 = 25 64 ? 3 8 ? 5 8
3 8 × × = 15 32 ?

8. Question 1 ?
1 5 × 1 5 = 1 25
1 5 × × 1 5 = 8 25 ?
1− 8 25 = 17 25 ?

9. Question 2 ?
0.9 ?
0.9
0.1
0.9 ?
0.1
0.1 ?
0.9 2 =0.81 ?
2×0.1×0.9=0.18

10. Question 3 ?
4 13
5 14 ?
9 13
5 13 ?
9 14
8 13
9 14 × 8 13 = 36 91 ?
two consonants?
5 14 × × = 45 91 ?

11. Question 4
2 9 ? ?
3 10
7 9
3 9 ?
7 10
6 9
3 10 × × 3 9 = 7 30 ?
1− 7 30 = 23 30 ?

12. Frequency Trees
30 students were asked if they liked coffee.
20 of the students were girls.
6 boys liked coffee.
12 girls did not like coffee.
Like coffee 8 20
girls 12
Don’t like coffee 30 6
Like coffee
boys 10 4
Don’t like coffee

13. 56 students were asked if they watched tennis yesterday.
20 of the students are boys.
13 boys did not watch tennis.
17 girls watched tennis.
Did watch tennis 17 36
girls 19
Didn’t watch tennis 56 7
Did watch tennis
boys 20 13
Didn’t watch tennis

14. 85 students were asked what they did for lunch, 47 of these were female.
5 boys had a packed lunch.
21 girls had a school dinner.
13 girls had ‘other’.
Altogether 40 pupils had a school dinner. 13
Packed lunch 47
School dinner 21
female
other 13 85 5
Packed lunch
male 38
School dinner 19 14
other

15. 18 47 14 15 80 8 33 19 6 80 Year 12 students each study one Science.
47 of these students are females.
18 girls study Biology, and 19 boys study physics.
Altogether 33 study physics and 21 pupils study chemistry. 18
Biology 47
Physics 14
female
chemistry 15 80 8
Biology
male 33
Physics 19 6
Chemistry