Frequency trees and two-way tables
Frequency trees
A shows how a group splits up, with a number on each branch.
Example: of 80 students, 48 have school lunch. Of these, 30 are in Year 10. Of the rest, 12 are in Year 10. Complete the tree and find P(a student chosen at random is in Year 11 with packed lunch).
- 80 students
- 48 school lunch: 30 Year 10, 18 Year 11
- 32 packed lunch: 12 Year 10, 20 Year 11
- School lunch, Year 11: 48 − 30 = 18
- Packed lunch: 80 − 48 = 32, and Year 11: 32 − 12 = 20
- P(Year 11 and packed lunch) = 20/80 = ¼
Check: each pair of branches adds up to the number before it.
Key terms
Two-way tables
A sorts data by two things at once. Each row and column adds up to its total.
Example: 90 students were asked about their pets. Fill in the gaps, then find P(a student chosen at random is a girl with a dog).
| Cat | Dog | Fish | Total | |
|---|---|---|---|---|
| Boys | 12 | ? | 5 | 40 |
| Girls | ? | 25 | 7 | ? |
| Total | 30 | ? | ? | 90 |
- Boys with a dog: 40 − 12 − 5 = 23
- Girls with a cat: 30 − 12 = 18
- Girls total: 90 − 40 = 50 (check: 18 + 25 + 7 = 50)
- Dog total: 23 + 25 = 48; fish total: 5 + 7 = 12
- P(girl with a dog) = 25/90 = 5/18
Key terms
Conditional probability from a two-way table (Higher tier only)
A is the probability of something given that you already know something else. Only the group you know about counts, so use its total.
Example: using the completed pets table, a student is chosen at random. Find P(girl, given the student has a dog) and P(cat, given the student is a girl).
- Given a dog: there are 48 dog owners, and 25 of them are girls
- P(girl given dog) = 25/48
- Given a girl: there are 50 girls, and 18 of them have a cat
- P(cat given girl) = 18/50 = 9/25
Common mistake
Use the total of the group you're given, not the grand total: 25/48, not 25/90.