Frequency tables and frequency trees
Probability is built on frequencies — counts of outcomes from real or imagined experiments. Before you compute any probability, you need to organise the data correctly.
Frequency tables
A frequency table records how often each outcome occurs. The simplest ones have two columns: outcome and frequency.
| Number on dice | Frequency |
|---|---|
| 1 | 12 |
| 2 | 14 |
| 3 | 8 |
| 4 | 11 |
| 5 | 13 |
| 6 | 12 |
Total throws = 70. Relative frequency of "rolled a 3" = 8/70 = 4/35.
Two-way (contingency) tables
Two-way tables compare two categorical variables.
| Walk | Bus | Cycle | Total | |
|---|---|---|---|---|
| Y10 | 12 | 18 | 5 | 35 |
| Y11 | 9 | 22 | 4 | 35 |
| Total | 21 | 40 | 9 | 70 |
Useful probabilities to extract:
- P(walks) = 21/70 = 3/10.
- P(Y11 and bus) = 22/70 = 11/35.
- P(walks given Y10) = 12/35 (conditional, see P9).
Always check that all rows sum to the row total and all columns sum to the column total. Examiners often leave gaps to fill in.
Frequency trees
A frequency tree is a diagram (similar to a probability tree) where the numbers at each branch are FREQUENCIES, not probabilities. They make conditional counting visual and easy.
Example: 80 students take an exam. 50 are female, 30 are male. Of the females, 35 pass; of the males, 22 pass.
80
/ \
Female Male
(50) (30)
/ \ / \
Pass Fail Pass Fail
(35) (15) (22) (8)
From the tree:
- P(pass) = (35 + 22)/80 = 57/80.
- P(female and pass) = 35/80 = 7/16.
Reading and completing trees
A typical question gives you partial frequencies and asks you to complete the rest. Always start with the totals you know and work outwards.
Worked example. 200 people are surveyed about voting. 120 will vote. Of those, 70 are women. Of the 80 not voting, 50 are men. Complete the tree.
- 120 voting → 70 women, so 120 − 70 = 50 men voting.
- 80 not voting → 50 men, so 80 − 50 = 30 women not voting.
- Total women = 70 + 30 = 100; total men = 50 + 50 = 100. ✓
Probability from frequency
For an outcome A based on N trials with frequency fA:
P(A) ≈ f(A) / N — the relative frequency estimate.
The more trials, the closer this gets to the true (theoretical) probability — see P5.
⚠Common mistakes— Common mistakes (examiner traps)
- Forgetting to sum the column total when finding marginal probabilities from a two-way table.
- Confusing "AND" and "GIVEN". P(A and B) divides by total; P(A | B) divides by the size of B.
- Reporting raw counts as probabilities. A probability is always between 0 and 1.
- Misreading the tree. The numbers at the second level are frequencies of "this outcome AND that previous branch", not totals.
- Not checking sums. A frequency tree's leaves should add up to the trunk total.
➜Try this— Quick check
A class of 30 has 18 girls and 12 boys. 13 girls play sport; 7 boys do not. Build a frequency tree and find the probability that a randomly chosen pupil is a boy who plays sport.
- Boys playing = 12 − 7 = 5.
- Girls not playing = 18 − 13 = 5.
- P(boy and sport) = 5/30 = 1/6.
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