P1: Recording Outcomes
Record, describe and analyse the frequency of outcomes of simple probability experiments
📋 Key Concepts
Definition: An outcome is a possible result of an experiment. Recording outcomes means systematically noting what happens each time an experiment is repeated.
Key Terms
| Term | Meaning | Example |
|---|---|---|
| Experiment | A process that produces outcomes | Rolling a dice |
| Outcome | A possible result | Getting a 6 |
| Event | A collection of outcomes | Getting an even number |
| Frequency | How often an outcome occurs | 6 appeared 4 times |
| Trial | One run of an experiment | One coin flip |
Methods of Recording
- Tally Chart: Use | || ||| |||| ||||| to count
- Frequency Table: Outcomes with counts
- Frequency Tree: Branching diagram
📝 Tally Charts
Method: Make one mark for each outcome. Group marks in fives (four vertical lines crossed by a diagonal).
Example 1
A dice is rolled 20 times. Record the outcomes using a tally chart.
Solution:
| Outcome | Tally | Frequency |
|---|---|---|
| 1 | |||| | 4 |
| 2 | ||| | 3 |
| 3 | |||| | 4 |
| 4 | || | 2 |
| 5 | |||| | | 6 |
| 6 | | | 1 |
Total: 4 + 3 + 4 + 2 + 6 + 1 = 20 trials
📝 Frequency Tables
Structure: List all possible outcomes in one column and their frequencies in another. You can add columns for relative frequency and cumulative frequency.
Example 2
A coin is flipped 50 times. Complete the frequency table.
Results: 23 heads, 27 tails
| Outcome | Frequency | Relative Frequency |
|---|---|---|
| Heads | 23 | 23/50 = 0.46 |
| Tails | 27 | 27/50 = 0.54 |
| Total | 50 | 1.00 |
Relative frequency = Frequency ÷ Total trials
📝 Frequency Trees
Use: Frequency trees show how a total is split into different categories. They are useful for two-stage experiments.
Example 3
80 students were asked if they walk or get the bus to school. 45 walk. Of those who walk, 30 are girls. Of those who get the bus, 20 are boys. Complete the frequency tree.
Solution:
Total: 80
/ \
Walk: 45 Bus: 35
/ \ / \
Girls Boys Girls Boys
30 15 15 20
Check: 30 + 15 + 15 + 20 = 80 ✓
📝 Two-Way Tables
Structure: Two-way tables (or contingency tables) show data organised by two categories.
Example 4
60 people were surveyed about their favourite sport. Results are shown below:
| Football | Rugby | Tennis | Total | |
|---|---|---|---|---|
| Male | 18 | 12 | 5 | 35 |
| Female | 8 | 7 | 10 | 25 |
| Total | 26 | 19 | 15 | 60 |
Each row and column can be used to find probabilities.
❓ Practice Questions
Q1: A dice is rolled 30 times. The tally for 4 shows |||| |||| |. What is the frequency?
Q2: In 100 trials, an event occurred 25 times. What is the relative frequency?
Q3: A bag contains coloured counters. 120 are drawn and recorded: 48 red, 72 blue. What fraction are blue?
Q4: Complete this frequency tree: 200 people, 130 are right-handed. Of right-handed, 70 are male. Of left-handed, 40 are female.
Q5: In a two-way table, 40 students like maths, 60 don't. Of those who like maths, 25 are girls. How many boys like maths?
✅ Answers
- 11 (each group of 5 + 1)
- 25/100 = 0.25
- 72/120 = 3/5 or 0.6
- Right-handed: 70 male, 60 female; Left-handed: 30 male, 40 female. Check: 70+60+30+40=200
- 40 - 25 = 15 boys like maths
🎯 Exam Tips
- Always check that frequencies add up to the total
- Use tally marks for data collection - it's harder to lose count
- Relative frequency is always between 0 and 1
- Check totals in two-way tables by adding rows AND columns
- Frequency trees are useful for conditional probability questions
🧠 Problem-Solving Strategies
Problem-Solving
When recording outcomes: (1) Always identify all possible outcomes before counting, (2) Use systematic methods (tally, tables, trees) to avoid missing outcomes, (3) Check that your totals add up — if they don't, you've missed or double-counted something, (4) For two-way tables, fill in what you know first then use row/column totals to find missing values.
Multi-Step Problem
In a survey of 120 students, each student studies exactly one language: French or Spanish. 70 are girls. 45 study French. 30 girls study Spanish. How many boys study French?
Solution: Total Spanish = 120 − 45 = 75. Girls studying Spanish = 30, so boys studying Spanish = 75 − 30 = 45. Total boys = 120 − 70 = 50. Boys studying French = 50 − 45 = 5. Check: French total = girls French + boys French = (70−30) + 5 = 40 + 5 = 45 ✓
⚠️ Common Errors
Watch Out!
1. Wrong: Forgetting to include all categories in a frequency table total Correct: Always check that the sum of all frequencies equals the total number of trials
2. Wrong: Counting a tally group of 5 as 4 because the diagonal line looks separate Correct: A diagonal line through four vertical lines makes a group of 5
3. Wrong: In a two-way table, adding the row totals and column totals separately without checking they match Correct: Row grand total must equal column grand total — if they differ, a cell is wrong
✍️ 6-Mark Exam Question
Extended Answer
6 marks: 80 people were surveyed about their pet ownership. 50 own a cat, 45 own a dog, and 15 own neither. Draw a two-way table and use it to find the probability that a randomly chosen person owns both a cat and a dog.
Step 1: People with pets = 80 − 15 = 65.
Step 2: Let x = both cat and dog. Then Cat only = 50 − x, Dog only = 45 − x.
Step 3: Cat only + Both + Dog only = 65, so (50−x) + x + (45−x) = 65, giving 95 − x = 65, so x = 30.
Step 4: Two-way table: Cat only = 20, Both = 30, Dog only = 15, Neither = 15.
Step 5: P(both cat and dog) = 30/80 = 3/8 = 0.375.
Mark scheme: M1 for finding 65 with pets, M1 for setting up equation with x, M1 for solving x = 30, M1 for correct two-way table, M1 for probability fraction, A1 for final answer 3/8
📊 AO3: Reason & Interpret
Reasoning and Interpretation
A school records how Year 10 and Year 11 students travel to school: Bus, Walk, or Cycle. The two-way table shows the results.
| Bus | Walk | Cycle | Total | |
|---|---|---|---|---|
| Year 10 | 25 | 30 | 15 | 70 |
| Year 11 | 20 | 35 | 25 | 80 |
| Total | 45 | 65 | 40 | 150 |
(a) What is the probability a randomly chosen student cycles?
(b) A student is chosen at random from those who walk. What is the probability they are in Year 11?
(c) Mia says "Year 11 students are more likely to cycle than Year 10 students." Is she correct? Show working.
Answers: (a) 40/150 = 4/15 ≈ 0.267 (b) 35/65 = 7/13 ≈ 0.538 (c) Yes — P(cycle|Year 10) = 15/70 ≈ 0.214, P(cycle|Year 11) = 25/80 ≈ 0.313. Year 11 proportion is higher.
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