What is probability?
Probability measures how likely an event is to happen, on a scale from 0 (impossible) to 1 (certain). It can be written as a fraction, decimal or percentage.
The key rules:
- P(event) = number of favourable outcomes ÷ total number of possible outcomes (when all outcomes are equally likely)
- All the probabilities for every possible outcome of an experiment must add up to 1
- P(not A) = 1 − P(A) — the probability an event does not happen
- For mutually exclusive events (they cannot happen at the same time), P(A or B) = P(A) + P(B) — this is the addition rule
The probability scale runs from 0 (impossible) to 1 (certain). Every probability, whether a fraction, decimal or percentage, sits somewhere on this line.
Calculating probability from equally likely outcomes
Worked Example 1
A bag contains 5 red counters, 3 blue counters and 2 green counters. A counter is picked at random. Find the probability it is blue.
1
Find the total number of counters: 5 + 3 + 2 = 10
2
Number of favourable outcomes (blue) = 3
3
P(blue) = 3 ÷ 10 = 3/10
Answer3/10 (or 0.3)
The "not" rule
Worked Example 2
The probability that it rains tomorrow is 0.35. Find the probability that it does not rain.
1
Use P(not A) = 1 − P(A)
2
P(not rain) = 1 − 0.35 = 0.65
Answer0.65
The addition rule
Worked Example 3
A spinner lands on red, blue, green or yellow. The table shows the probability of each colour. Find P(red or yellow).
| Colour | Red | Blue | Green | Yellow |
|---|---|---|---|---|
| Probability | 0.2 | 0.35 | 0.15 | 0.3 |
1
Red and yellow are mutually exclusive (the spinner cannot land on both), so use the addition rule
2
P(red or yellow) = P(red) + P(yellow) = 0.2 + 0.3 = 0.5
Answer0.5
Relative frequency and expected frequency
When outcomes are not equally likely, or when you're testing whether they are, probability can be estimated from experimental results using relative frequency = number of times the event happens ÷ total number of trials. This estimate can then be used to find an expected frequency for a larger number of trials: expected frequency = probability × number of trials.
Worked Example 4 — Relative frequency
A biased dice is rolled 200 times. The table shows the results. Estimate the probability of rolling a 3, and use it to predict how many 3s would occur in 500 rolls.
| Score | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|
| Frequency | 28 | 30 | 62 | 26 | 25 | 29 |
1
Relative frequency of rolling a 3 = 62 ÷ 200 = 0.31
2
Expected frequency in 500 rolls = 0.31 × 500 = 155
AnswerP(3) ≈ 0.31; expected 155 rolls out of 500
Practice questions
Work through each question before checking the answers.
Foundation (Grade 3–5)
Q1A bag contains 4 red counters and 6 blue counters. A counter is picked at random. Find P(red).Foundation
Q2A fair six-sided dice is rolled once. Find the probability of scoring more than 4.Foundation
Q3The probability that a bus arrives late is 0.15. Find the probability that it does not arrive late.Foundation
Q4A fair spinner has 8 equal sections numbered 1 to 8. Find the probability of landing on an even number.Foundation
Q5A box contains 12 chocolates: 5 milk, 4 dark and 3 white. One is chosen at random. Find P(dark or white).Foundation
Higher (Grade 5–7)
Q6
100 cars pass a junction. The table shows their colour. Find the relative frequency of a car being silver.
Higher| Colour | Red | Blue | Silver | Black |
|---|---|---|---|---|
| Frequency | 18 | 25 | 31 | 26 |
Q7Using the data from Q6, predict how many of the next 250 cars will be blue.Higher
Q8A bag contains only red and blue counters. P(red) = 0.4. There are 30 counters in total. How many are blue?Higher
Q9Events A and B are mutually exclusive. P(A) = 0.3 and P(A or B) = 0.55. Find P(B).Higher
Q10A spinner lands on red, blue or green. P(red) = 0.25 and P(blue) = 0.4. Find P(green).Higher
Higher — Hard (Grade 8–9)
Q11A biased coin has P(heads) = 0.6. It is flipped 150 times. Find the expected number of tails.Grade 8–9
Q12A spinner is spun 300 times and lands on red 84 times. Estimate P(red) as a fraction in its simplest form, then predict the number of reds in 1000 spins.Grade 8–9
Q13Three mutually exclusive events A, B and C have probabilities P(A) = 2x, P(B) = x, P(C) = 3x. Find the value of x.Grade 8–9
Q14A four-sided spinner has P(1) = 0.2, P(2) = 0.3, P(3) = y, P(4) = 2y. Find the value of y.Grade 8–9
Q15A dice is rolled 400 times and lands on 6 a total of 96 times. State, with a reason based on relative frequency, whether this suggests the dice is biased.Grade 8–9
Answers
Foundation (Q1–Q5)
Q12/5(4 ÷ 10, simplified)
Q21/3(scores of 5 or 6 → 2 ÷ 6)
Q30.85(1 − 0.15)
Q41/2(4 even numbers ÷ 8)
Q57/12(P(dark) + P(white) = 4/12 + 3/12)
Higher (Q6–Q10)
Q60.31(31 ÷ 100)
Q7≈ 63 cars(P(blue) = 0.25; 0.25 × 250 = 62.5)
Q818(P(blue) = 0.6; 0.6 × 30)
Q90.25(0.55 − 0.3)
Q100.35(1 − 0.25 − 0.4)
Higher — Hard (Q11–Q15)
Q1160(P(tails) = 0.4; 0.4 × 150)
Q127/25; 280 reds(84 ÷ 300 = 0.28 = 7/25; 0.28 × 1000)
Q13x = 1/6(2x + x + 3x = 6x = 1)
Q14y = 1/6 (≈0.167)(0.5 + 3y = 1)
Q15Relative frequency = 96 ÷ 400 = 0.24, compared with the theoretical probability for a fair dice of 1/6 ≈ 0.167. Since 0.24 is noticeably higher, this suggests the dice may be biased towards rolling a 6 — though more trials would give a more reliable estimate.
Common mistakes
Common Mistake 1
Writing probability as a ratio instead of a fraction
P(red) for 4 red counters out of 10 total is 4/10, not 4:6 or "4 out of 6". Always compare the favourable outcomes to the total number of outcomes, not to the other outcomes.
Common Mistake 2
Using the addition rule on non-mutually exclusive events
P(A or B) = P(A) + P(B) only works when A and B cannot happen at the same time. If events overlap (e.g. "even number" and "greater than 3" on a dice both include 4 and 6), this simple addition would double-count and give the wrong answer.
Common Mistake 3
Forgetting that all probabilities must sum to 1
If a question gives probabilities for every possible outcome, they must add up to exactly 1. Use this to check your answer, or to find a missing probability by subtracting the others from 1.
Common Mistake 4
Treating expected frequency as a guaranteed outcome
Expected frequency is a prediction based on probability, not a certainty. If P(heads) = 0.5 and a coin is flipped 100 times, you would expect 50 heads, but getting 47 or 53 does not mean the calculation was wrong.
Exam tips
💡 Exam Tip 1
Leave answers as fractions unless told otherwise
Unless the question specifically asks for a decimal or percentage, a simplified fraction (e.g. 2/5 rather than 0.4) is usually the safest and clearest way to present a probability answer.
💡 Exam Tip 2
Use the "sum to 1" check
Before submitting an answer involving several probabilities, add them all up. If they don't sum to 1 (for a complete set of outcomes), you've made an error somewhere — go back and check.
💡 Exam Tip 3
Distinguish theoretical from experimental probability
If a question gives you results from trials (relative frequency), don't switch to counting outcomes as if they were all equally likely. State clearly which type of probability you are using in your answer.
💡 Exam Tip 4
Define events clearly in worded answers
For questions asking you to explain or justify (like judging whether a dice is biased), name the specific numbers you're comparing — the experimental relative frequency versus the theoretical probability — rather than just saying "it seems biased".
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