P3: Probability Scale
Understand the 0-1 probability scale; use relative frequency and theoretical probability
📋 Key Concepts
The Probability Scale: All probabilities are between 0 and 1. 0 means impossible, 1 means certain, 0.5 means even chance.
The Probability Scale
0 0.25 0.5 0.75 1
|---------|----------|----------|----------|
Impossible Unlikely Even Likely Certain
1/4 Chance 3/4
50%
Key Probabilities
| Value | Meaning | Example |
|---|---|---|
| 0 | Impossible (will never happen) | Rolling a 7 on normal dice |
| 0.5 | Even chance (50-50) | Getting heads on fair coin |
| 1 | Certain (will definitely happen) | Rolling less than 7 on normal dice |
📝 Theoretical Probability
Formula: P(event) = Number of favourable outcomes ÷ Total number of possible outcomes
P(event) = Number of ways event can happen/Total number of possible outcomes
Example 1
A fair 6-sided dice is rolled. Find P(rolling an even number).
Solution:
Favourable outcomes: 2, 4, 6 (3 outcomes)
Total outcomes: 1, 2, 3, 4, 5, 6 (6 outcomes)
P(even) = 3/6 = 1/2 = 0.5
Example 2
A bag contains 5 red, 3 blue and 2 green balls. Find P(red).
Solution:
Total balls = 5 + 3 + 2 = 10
P(red) = 5/10 = 1/2 = 0.5
📝 Relative Frequency
Definition: Relative frequency is the experimental probability based on actual trials. It estimates the true probability.
Relative Frequency = Number of times event occurs/Total number of trials
Example 3
A coin is flipped 200 times. Heads appears 95 times. Find the relative frequency of heads.
Solution:
Relative frequency = 95/200 = 0.475
Note: Theoretical probability is 0.5, but experimental varies.
Example 4
A dice is rolled 60 times. The number 3 appears 8 times. Find the relative frequency of rolling a 3.
Solution:
Relative frequency = 8/60 = 2/15 ≈ 0.133
Theoretical probability would be 1/6 ≈ 0.167
📝 Words and Probability
Language: Probability can be described using words: impossible, unlikely, even chance, likely, certain.
Example 5
Describe the probability of each event using words from the probability scale:
a) P(snow in July in UK) ≈ 0.05
b) P(getting a head when flipping a coin) = 0.5
c) P(getting a number less than 7 on a dice) = 1
Answers:
a) Very unlikely (close to 0)
b) Even chance (exactly 0.5)
c) Certain (exactly 1)
📝 Comparing Probabilities
Tip: Convert all probabilities to the same format (decimals, fractions or percentages) to compare them easily.
Example 6
Which is more likely: P(A) = 3/8 or P(B) = 0.35?
Solution:
P(A) = 3/8 = 0.375
P(B) = 0.35
0.375 > 0.35, so P(A) is more likely
🧮 Worked Examples — Exam Style
Probability sits between 0 (impossible) and 1 (certain). Theoretical = favourable ÷ total; relative frequency = observed ÷ trials.
A spinner has 5 equal sections; 2 are shaded. Find P(not shaded) as a fraction and a decimal.
P(shaded) = 2/5 → P(not shaded) = 3/5 = 0.6
B1 for 3/5 = 0.6. 'Not' event: 1 − P(event). Fractions and decimals both accepted.
A die is rolled 200 times; six appears 38 times. Compare with the theoretical probability.
Theoretical P(6) = 1/6 ≈ 0.167. Relative frequency = 38/200 = 0.19
Close to theory — consistent with a fair die (small sample variation expected).
M1 for both probabilities, A1 for the comparison with reasoning. 'Compare' needs BOTH numbers.
Exam wording to expect: "Find the probability of …", "mark on the probability scale", "compare with the theoretical probability".
❓ Practice Questions
Q1: A fair dice is rolled. Find P(rolling a number greater than 4).
Q2: A bag has 4 red, 5 blue and 1 green ball. What is P(green)?
Q3: A spinner is spun 80 times. It lands on red 32 times. Find the relative frequency of red.
Q4: Place these on a probability scale: P(A) = 0, P(B) = 2/3, P(C) = 0.25, P(D) = 1
Q5: A coin is flipped 1000 times. Heads appears 485 times. Is the coin fair?
Q6: A bag holds 3 red, 5 blue, 2 green. One drawn at random: P(not red)?
Q7: P(rain) = 0.7. What is P(no rain), and what does it mean?
✅ Answers
- P(5 or 6) = 2/6 = 1/3
- P(green) = 1/10 = 0.1
- Relative frequency = 32/80 = 0.4
- P(A) = 0 (impossible), P(C) = 0.25 (unlikely), P(B) = 2/3 ≈ 0.67 (likely), P(D) = 1 (certain)
- Relative frequency = 485/1000 = 0.485. This is close to 0.5, so the coin appears fair (within expected variation).
📋 Model Answers — Full Working
Every mark comes from the working, not just the answer. These model answers show the lines a mark scheme expects to see.
Q1. P(5 or 6) = 2/6 = 1/3
Q2. 1/10 = 0.1
Q3. 32/80 = 0.4
Q4. 0 at the left end (impossible), 0.25 a quarter along (unlikely), 2/3 two-thirds along (likely), 1 at the right end (certain)
Q5. 485/1000 = 0.485 ≈ 0.5 → consistent with a fair coin — small deviation is expected from randomness
Q6. P(red) = 3/10 → P(not red) = 7/10
Q7. P(no rain) = 1 − 0.7 = 0.3 — a 30% chance, closer to ‘unlikely’ than ‘likely’ on the scale
📌 Key Facts to Memorise
- Scale: 0 impossible → 0.25 unlikely → 0.5 even chance → 0.75 likely → 1 certain
- Theoretical P = favourable outcomes ÷ total outcomes (equally likely)
- Relative (experimental) frequency = times it happened ÷ trials
- P(not A) = 1 − P(A) · all probabilities together sum to 1
- More trials → relative frequency converges to the theoretical probability
🗺️ In the Exam
Probability basics: 2–4 marks on every paper. Scale-marking and relative-frequency-vs-theory comparisons are the AO2/AO3 staples.
Links to: possibility spaces (P7), experimental probability (P5), listing (N5).
🧭 Board Focus — AQA
Spec 8300. Three papers, each 80 marks and 1h30: Paper 1 non-calculator, Papers 2 and 3 calculator. Foundation/Higher tiers. Roughly a third of the marks sit under AO2 (reasoning) and AO3 (problem-solving), so show every line of working — method marks survive arithmetic slips.
Always check: specification details change — confirm paper structure and set texts against the current specification for your exam year.
🎯 Exam Tips
- Always give probabilities as fractions, decimals or percentages
- Never give probability as a ratio or "1 in 6" - use 1/6 instead
- Theoretical probability is calculated; relative frequency is from experiments
- More trials = relative frequency gets closer to theoretical probability
- Check your answer is between 0 and 1 - if not, you've made an error
🧠 Problem-Solving Strategies
Problem-Solving
When placing events on the probability scale: (1) Calculate the exact probability first, (2) Convert to a decimal to locate it on the 0–1 scale, (3) Use the words impossible/unlikely/even chance/likely/certain to describe position, (4) For comparison problems, always convert to the same format (decimals, fractions, or percentages).
Multi-Step Problem
A bag contains 3 red, 5 blue and 2 green counters. A counter is drawn at random. Place each event on the probability scale: (a) P(red), (b) P(not blue), (c) P(red or green).
Solution: Total = 10. (a) P(red) = 3/10 = 0.3 — unlikely. (b) P(not blue) = 1 − 5/10 = 0.5 — even chance. (c) P(red or green) = (3+2)/10 = 5/10 = 0.5 — even chance.
⚠️ Common Errors
Watch Out!
1. Wrong: Giving probability as a ratio like "1 in 6" or "1:6" Correct: Write as a fraction 1/6, decimal 0.167, or percentage 16.7%
2. Wrong: Writing a probability greater than 1 or less than 0 Correct: All probabilities must be between 0 and 1 inclusive — if your answer is outside this range, check your calculation
3. Wrong: Confusing relative frequency with theoretical probability when asked to "calculate" probability Correct: "Calculate" means use theoretical probability (favourable/total); "estimate from experiment" means use relative frequency
✍️ 6-Mark Exam Question
Extended Answer
6 marks: A fair 10-sided dice has faces numbered 1 to 10. (a) Calculate P(prime number). (b) Calculate P(multiple of 3). (c) Show that P(prime number or multiple of 3) is NOT found by simply adding your answers to (a) and (b). Explain why.
(a) Primes from 1–10: 2, 3, 5, 7 → 4 primes. P(prime) = 4/10 = 2/5 = 0.4.
(b) Multiples of 3 from 1–10: 3, 6, 9 → 3 values. P(multiple of 3) = 3/10 = 0.3.
(c) Numbers that are prime OR multiple of 3: 2, 3, 5, 6, 7, 9 → 6 values. P(prime or multiple of 3) = 6/10 = 0.6. But 0.4 + 0.3 = 0.7 ≠ 0.6. Simply adding gives the wrong answer because 3 is counted twice — it is both prime AND a multiple of 3. The events are not mutually exclusive.
Mark scheme: M1 for listing primes, A1 for P(prime)=0.4, M1 for listing multiples of 3, A1 for P(multiple of 3)=0.3, M1 for demonstrating overlap, A1 for explanation about non-mutually-exclusive events
📊 AO3: Reason & Interpret
Reasoning and Interpretation
A weather app says there is a 70% chance of rain tomorrow. Sam says "It will definitely rain." Nina says "It might rain."
(a) On the probability scale, where does 70% sit?
(b) Who is correct — Sam or Nina? Explain your answer.
(c) Over 100 days with the same forecast, on how many days would you expect it to rain? Is this a guarantee?
Answers: (a) 0.7 — between "likely" and "certain", closer to "likely". (b) Nina — 0.7 means likely but not certain (not equal to 1). Sam is wrong because 70% ≠ 100%. (c) Expected = 70 out of 100 days. Not a guarantee — probability is about expectation, not certainty for any single day.
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🎬 Video Resources
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