IGCSE Maths · Probability · Core & Extended

Probability: The Scale, Theoretical Probability and Frequency

This topic is identical for Core and Extended — every idea here is examined the same way at both tiers. Probability puts a number on uncertainty: 0 means “never happens,” 1 means “always happens,” and everything else sits somewhere in between.

1. The Probability Scale and Notation

The scale from 0 to 1 0 0.5 1 impossible unlikely evens likely certain Probabilities are written as fractions, decimals, or percentages between 0 and 1 — never negative, and never more than 1.

Notation: P(A) means “the probability that event A happens.”

2. Theoretical Probability of a Single Event

P(event) = (number of favourable outcomes) ÷ (total number of possible outcomes)
Worked examples A fair six-sided die is rolled. Find P(rolling a 4).
There’s 1 favourable outcome (rolling a 4) out of 6 possible outcomes: P(4) = 1/6

Find P(rolling an even number).
Favourable outcomes: 2, 4, 6 — that’s 3 out of 6: P(even) = 3/6 = 1/2

A bag contains 5 red, 4 blue, and 3 green balls (12 total). Find P(red).
P(red) = 5/12
Try it — Theoretical Probability
  1. Find P(rolling a number greater than 4) on a fair die.
  2. A bag has 6 yellow, 3 purple, and 1 white counter. Find P(purple).
  3. A spinner has 8 equal sections, numbered 1–8. Find P(spinning a multiple of 3).
Show answers
  1. Favourable: 5, 6 → 2/6 = 1/3
  2. 3/10
  3. Multiples of 3 up to 8: 3, 6 → 2/8 = 1/4

3. The Probability of an Event Not Happening

P(not A) = 1 − P(A)
Worked examples Find P(not rolling a 6) on a fair die.
P(6) = 1/6, so P(not 6) = 1 − 1/6 = 5/6

Using the ball bag from before (5 red out of 12), find P(not red).
P(not red) = 1 − 5/12 = 7/12
Try it — P(not A)
  1. P(rain tomorrow) = 0.35. Find P(no rain tomorrow).
  2. A bag has 6 yellow, 3 purple, 1 white counter (10 total). Find P(not yellow).
Show answers
  1. 1 − 0.35 = 0.65
  2. 1 − 6/10 = 4/10 = 2/5

4. Relative Frequency as an Estimate of Probability

When the true probability isn’t known (or to check whether a die/coin is fair), repeat an experiment many times and use the results as an estimate.

Relative frequency = (number of times event occurred) ÷ (total number of trials)
Worked example A coin is flipped 200 times and lands on heads 112 times. Find the relative frequency of heads, and use it to comment on whether the coin seems fair.
Relative frequency = 112/200 = 0.56
This is close to the theoretical probability of 0.5 for a fair coin, so there’s no strong evidence the coin is biased — small differences from 0.5 are expected just from natural variation.
Try it — Relative Frequency
  1. A die is rolled 150 times and lands on 6 a total of 31 times. Find the relative frequency of rolling a 6.
  2. Compare your answer to the theoretical probability of 1/6 (≈0.167). Does the die seem fair?
Show answers
  1. 31/150 = 0.207 (3 d.p.)
  2. 0.207 is a bit higher than 0.167, but with only 150 trials this could still just be natural variation — more trials would give a more reliable estimate before concluding the die is biased.

5. Expected Frequency

If you know (or estimate) a probability, you can predict how many times an event should happen over a number of trials.

Expected frequency = P(event) × number of trials
Worked examples A spinner has P(landing on red) = 0.3. It’s spun 150 times. Find the expected number of reds.
Expected frequency = 0.3 × 150 = 45

A fair die is rolled 300 times. Find the expected number of 6s.
Expected frequency = (1/6) × 300 = 50
Try it — Expected Frequency
  1. P(a spinner landing on blue) = 0.25. It’s spun 80 times. Find the expected number of blues.
  2. A fair die is rolled 240 times. Find the expected number of times an even number is rolled.
  3. A factory finds 2% of items are faulty. In a batch of 3,500 items, find the expected number of faulty ones.
Show answers
  1. 0.25×80 = 20
  2. 0.5×240 = 120
  3. 0.02×3500 = 70
Common mistakes to watch for
  • Writing a probability greater than 1, or negative — always sanity-check that your answer sits between 0 and 1.
  • Forgetting relative frequency is only an estimate — it usually won’t exactly match the true theoretical probability, especially with a small number of trials.
  • Mixing up relative frequency (a result from experiments) with expected frequency (a prediction using a known/assumed probability) — they answer different questions.
  • Forgetting that P(not A) = 1 − P(A) only works when A is the complete opposite of “not A” — every outcome is either A or not A, with nothing missing.
  • Rounding expected frequency to a non-whole number when the context requires a whole number of events (e.g. “48.6 faulty items” should usually be interpreted/rounded sensibly depending on what’s being asked).

6. Quick-Fire Challenge Round

True or False?
  1. A probability of 0 means an event is certain to happen.
  2. P(not A) = 1 − P(A).
  3. Relative frequency always exactly equals the true probability.
  4. Expected frequency can be calculated without knowing (or estimating) a probability.
Show answers
  1. False — a probability of 0 means the event is impossible; 1 means certain.
  2. True.
  3. False — relative frequency is only an estimate, and usually differs slightly from the true probability, especially with fewer trials.
  4. False — you need a probability (known or estimated) to calculate an expected frequency.
Spot the Error
  1. “P(rolling a 6 or lower) on a fair die = 7/6.”
  2. “A coin is flipped 50 times, landing heads 40 times, so P(heads) is definitely 0.8 from now on.”
  3. “P(rain) = 0.4, so P(no rain) = 0.4 as well.”
Show answers
  1. A probability can never exceed 1 — rolling a 6 or lower on a standard die is certain, so P = 1, not 7/6 (a die only has faces 1–6, so “6 or lower” covers every outcome).
  2. This confuses relative frequency (an estimate from one set of experimental results) with a fixed, permanent probability — 40/50 = 0.8 is just this particular sample’s result, not a proven exact probability for all future flips.
  3. P(no rain) = 1 − 0.4 = 0.6, not 0.4 — the two probabilities must add to 1, not be equal.

7. Mixed Practice — Bringing It All Together

  1. A bag has 7 red, 5 blue, 8 green counters. Find P(blue).
  2. Using the same bag, find P(not green).
  3. A spinner lands on red with relative frequency 0.42 after 250 spins. How many times did it land on red?
  4. P(a machine produces a faulty item) = 0.015. In a run of 2000 items, find the expected number of faulty items.
Show answers
  1. 5/20 = 1/4
  2. 1 − 8/20 = 12/20 = 3/5
  3. 0.42×250 = 105 times
  4. 0.015×2000 = 30

Practice at Home

A large set to open the Probability unit — take it steadily.

Part A — Theoretical probability
  1. Find P(rolling a 2 or a 5) on a fair die.
  2. A bag has 9 orange, 6 green, 5 pink balls. Find P(pink).
  3. A spinner has 10 equal sections numbered 1–10. Find P(spinning a factor of 12).
Part B — P(not A)
  1. P(a train being late) = 0.18. Find P(the train being on time).
  2. Using the ball bag from Part A, find P(not orange).
Part C — Relative frequency
  1. A spinner is spun 400 times and lands on green 96 times. Find the relative frequency of green.
  2. A die is rolled 500 times and lands on an even number 240 times. Find the relative frequency, and compare to the theoretical probability of 0.5.
Part D — Expected frequency
  1. P(a spinner landing on yellow) = 0.4. It’s spun 120 times. Find the expected number of yellows.
  2. A fair coin is flipped 350 times. Find the expected number of tails.
  3. A shop finds 3% of returns are due to a manufacturing fault. Out of 1,200 returns, find the expected number due to a manufacturing fault.
Part E — Applied / challenge
  1. A bag contains only red and blue balls. P(red) = 0.35. If there are 140 balls in total, find the number of red balls and the number of blue balls.
  2. A weather app says P(rain today) = 0.7. Explain, in your own words, what this number does and doesn’t tell you about whether it will actually rain today.
  3. Over 5 days, a spinner is spun 100 times each day, landing on red 18, 22, 19, 21, and 20 times respectively. Find the overall relative frequency of red across all 5 days combined, and explain why this combined estimate is likely more reliable than any single day’s result.

Key Vocabulary Recap

Probability scale — runs from 0 (impossible) to 1 (certain)
Theoretical probability — calculated from known, equally likely outcomes
Relative frequency — an experimental estimate of probability from repeated trials
Expected frequency — a prediction of how many times an event should occur

Syllabus Reference

This note covers Cambridge IGCSE Mathematics (0580) sections C8.1 and C8.2 (Core subject content) and sections E8.1 and E8.2 (Extended subject content) — Introduction to probability, and Relative and expected frequencies. As with several Number-unit topics, the wording of these sections is identical for both tiers. Probability of combined events (sample space diagrams, Venn diagrams, tree diagrams, and Extended’s conditional probability) is covered in a separate note.

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