← IGCSE / O Level Mathematics

IGCSE / O Level Mathematics · Topic 8 Probability · Chapter 33 · pp. 476–487

Probability

Probability measures how likely something is, as a number from 0 (it cannot happen) to 1 (it must happen). Count the outcomes you want, divide by all the equally likely outcomes, and you have it. Then test the answer by experiment: the more trials you run, the closer the results get to the theory.

Learning objectives

  1. Place events on the probability scale from 0 to 1 and write probabilities as fractions, decimals and percentages.
  2. Calculate theoretical probability = favourable outcomes ÷ total outcomes.
  3. Use P(not A) = 1 − P(A).
  4. Find probabilities from tables, bar charts and Venn diagrams, including P(A ∩ B) and P(A ∪ B).
  5. Use relative frequency, and expected frequency = probability × number of trials.

Book exercises: 33.1 to 33.5 and Student assessments 1 and 2 (pp. 476–487).

The probability scale

0 ½ 1 impossible unlikely evens likely certain 0 = 0% · ½ = 0.5 = 50% · 1 = 100%
Every probability sits between 0 and 1. Evens (50-50) is halfway.

🔑 Key ideas

  • P = 0: the event is impossible. P = 1: it is certain.
  • A probability is never below 0 or above 1.
  • Write it as a fraction, a decimal or a percentage: = 0.25 = 25%.
  • If an event either happens or does not: P(not A) = 1 − P(A).

📘 Example 1 · Place “a day chosen at random is a Saturday” on the scale

  1. There are 7 days, all equally likely, and 1 of them is Saturday.
  2. P = ≈ 0.143 = 14.3%.
  3. That is between 0 and ½, close to 0, so mark it in the unlikely part of the scale.

📘 Example 2 · The sun will rise tomorrow, and a coin spun shows tails

  1. The sun will rise tomorrow: certain, so P = 1 (100%).
  2. A fair coin: 1 tail out of 2 faces, so P(tails) = = 0.5 = 50%, exactly at evens.

✏️ Try it

  1. The probability of rain is 0.3. What is the probability of no rain?
  2. Write 0.35 as a fraction and a percentage.
  3. Is it possible for P(A) to be 1.2? Explain.
Answers

1. 1 − 0.3 = 0.7 · 2. 35/100 = 7/20, and 35% · 3. No: a probability cannot be more than 1 (certain).

Next: theoretical probability →

Theoretical probability

🔑 The formula

P(event) =

  • Favourable outcome: the event you want actually happening.
  • The outcomes must be equally likely (a fair dice, an unbiased spinner, a random pick).
1 2 3 4 5 6 Even numbers (2, 4, 6) are the 3 favourable outcomes out of 6
The six outcomes of a fair dice. P(even) = 3 ÷ 6 = ½.

📘 Example 1 · A fair dice is rolled. Find P(six), P(not six) and P(even)

  1. Total outcomes = 6 (1, 2, 3, 4, 5, 6).
  2. P(six) = .
  3. P(not six) = 1 − = .
  4. Even numbers are 2, 4, 6: 3 favourable, so P(even) = = .

📘 Example 2 · A bag has 3 red, 10 white, 5 blue and 2 green counters

  1. Total counters = 3 + 10 + 5 + 2 = 20.
  2. P(green) = = .
  3. A green counter is taken and not put back, so 19 counters remain.
  4. P(second is red) = (all 3 reds are still there).
  5. P(second is green) = (only 1 green is left).

📘 Example 3 · A card is picked from a normal pack of 52. Find P(a 7 or a club)

  1. There are 4 sevens and 13 clubs.
  2. The 7 of clubs is in both lists, so do not count it twice: 4 + 13 − 1 = 16.
  3. P = = .

✏️ Try it

  1. A bag has 3 red and 7 blue counters. Find P(red).
  2. A card is picked from a pack of 52. Find P(a king).
  3. A letter is picked at random from MATHS. Find P(a vowel).
Answers

1. 3/10 · 2. 4/52 = 1/13 · 3. Only A is a vowel: 1/5

Next: tables and charts →

Probability from tables and charts

Read the totals from the table, then use favourable ÷ total. Always find the grand total first.

MathsEnglishArtTotal
Girls74516
Boys53614
Total1271130
Girls: 16 Boys: 14 Bar width is proportional to the count. Grand total = 16 + 14 = 30
The book's survey of 30 students. Girls 7 + 4 + 5 = 16, boys 5 + 3 + 6 = 14.

📘 Example 1 · A student is chosen. Use the table above

  1. Total students = 30.
  2. P(a girl) = = .
  3. P(a boy whose favourite is Art) = = .
  4. P(not a girl whose favourite is English): girls who like English = 4, so 1 − = = .
  5. Check: 7 + 5 + 5 + 3 + 6 = 26 students are outside that group. ✔

📘 Example 2 · Vehicles: Car 28, Lorry 6, Van 20, Bicycle 48, Motorbike 32, Other 6

  1. Total = 28 + 6 + 20 + 48 + 32 + 6 = 140.
  2. P(car) = = .
  3. P(not a van) = 1 − = = .

✏️ Try it

How pupils get to school. Boys: walk 8, bus 10, car 2. Girls: walk 6, bus 12, car 2.

  1. Find P(a girl).
  2. Find P(travels by bus).
  3. Find P(not a boy who walks).
Answers

Total = 20 + 20 = 40. 1. 20/40 = 1/2 · 2. 22/40 = 11/20 · 3. 1 − 8/40 = 32/40 = 4/5

Next: Venn diagrams →

Venn diagrams and probability

🔑 Set notation

  • ξ: the universal set (everything). { }: a set.
  • A ∪ B (union): in A or B or both.
  • A ∩ B (intersection): in A and B.
  • A′: not in A. P(A) = n(A) ÷ n(ξ).
ξ A B 3, 6, 9, 12 15 5, 10 outside both: 1, 2, 4, 7, 8, 11, 13, 14
Cards 1 to 15. A = multiples of 3, B = multiples of 5. 15 is the only card in both.

📘 Example 1 · Cards 1 to 15. A = multiples of 3, B = multiples of 5

  1. Multiples of 3: 3, 6, 9, 12, 15, so n(A) = 5. Multiples of 5: 5, 10, 15, so n(B) = 3.
  2. P(A) = = .
  3. Only 15 is in both, so P(A ∩ B) = .
  4. A ∪ B has 5 + 3 − 1 = 7 cards, so P(A ∪ B) = .
  5. P(A ∪ B)′ = 1 − = . (The 8 cards outside both circles.)

📘 Example 2 · 30 students study Spanish (S) or Mandarin (M). 16 only S, 8 only M

  1. Everyone studies at least one, so 16 + (both) + 8 = 30. Both = 30 − 24 = 6.
  2. P(S) = (16 + 6) ÷ 30 = = .
  3. P(M) = (8 + 6) ÷ 30 = = .
  4. P(S ∩ M) = = .
  5. P(S ∪ M) = = 1, since everyone is in at least one circle.

📘 Example 3 · 100 students: 70 track (T), 62 field (F), 8 neither

  1. In at least one = 100 − 8 = 92.
  2. Both: 70 + 62 − 92 = 40. Only T = 30, only F = 22.
  3. Check: 30 + 40 + 22 + 8 = 100. ✔
  4. P(T ∩ F) = = .
  5. P(F′) = 1 − = .

✏️ Try it

Of 30 students, 12 take Art (A), 15 take Music (M) and 5 take both.

  1. Find P(A ∩ M).
  2. Find P(A ∪ M).
  3. Find P(neither).
Answers

Only A = 7, only M = 10, both 5, so 22 in a circle and 8 outside. 1. 5/30 = 1/6 · 2. 22/30 = 11/15 · 3. 8/30 = 4/15

Next: relative frequency →

Relative frequency

🔑 Key ideas

  • Relative frequency = . It estimates probability from an experiment.
  • Expected frequency = P × number of trials. It is a prediction, not a promise.
  • More trials give a more reliable estimate: the relative frequency settles near the theoretical probability.
  • Rearranged: favourable = P × total, and total = favourable ÷ P.
P = 0.032 × 250 people = 8 she knows favourable = probability × total
The book's example: a girl knows 0.032 × 250 = 8 people in the hall.

📘 Example 1 · P(knowing someone) = 0.032 in a group of 250. How many does she know?

  1. Favourable = P × total = 0.032 × 250.
  2. 0.032 × 250 = 8 people.

📘 Example 2 · A boy enters 8 dogs. His chance of first prize is 0.016. How many dogs entered?

  1. P = favourable ÷ total, so total = favourable ÷ P.
  2. Total = 8 ÷ 0.016 = 500 dogs.
  3. Check: 8 ÷ 500 = 0.016. ✔

📘 Example 3 · A dice is rolled 60 times and 47 threes come up. Is it biased?

  1. Fair dice: expected number of threes = × 60 = 10.
  2. Relative frequency of 3 = ≈ 0.78, far above ≈ 0.17.
  3. Sixty rolls is a fair-sized sample, so this is strong evidence the dice is biased.
  4. A test of only 12 rolls (2 of each face) is too small to rely on.

🧪 Try it: probability experiment

1 0 number of trials P = ?
The solid line is the relative frequency as trials build up. The dashed line is the theoretical probability.

Working

    ✏️ Try it

    1. P(a faulty item) = 0.06. In a batch of 500, how many are likely to be faulty?
    2. 14 sweets are red and P(red) = 0.35. How many sweets are in the bag?
    3. A coin lands heads 90 times in 150 tosses. Find the relative frequency of heads.
    Answers

    1. 0.06 × 500 = 30 · 2. 14 ÷ 0.35 = 40 · 3. 90/150 = 0.6 (the coin may be biased; more tosses would tell)

    Next: summary and quiz →

    Chapter summary

    • Probability runs from 0 (impossible) to 1 (certain). P(not A) = 1 − P(A).
    • P(event) = favourable outcomes ÷ total equally likely outcomes. Simplify the fraction.
    • For tables, find the grand total first. Without replacement, the total goes down by one each time.
    • Venn diagrams: P(A) = n(A) ÷ n(ξ); A ∩ B is “and”, A ∪ B is “or”, A′ is “not”.
    • Relative frequency = times it happened ÷ trials. Expected frequency = P × n.
    • More trials make relative frequency a more reliable estimate of the true probability.

    📄 Practice questions

    The book's Exercises 33.1 to 33.5 and Student assessments 1 and 2 are in the printable assignment for this chapter.

    Mini-quiz

    1. A fair dice is rolled. What is the probability of getting a prime number?
    2. The probability that it rains tomorrow is 0.35. What is the probability that it does not rain?
    3. A bag holds 4 red, 6 blue and 10 green counters. What is P(not green)?
    4. Cards are numbered 1 to 20. What is the probability of picking a multiple of 5?
    5. A coin is spun 200 times and lands heads 94 times. What is the relative frequency of heads?
    6. The probability that an egg is bad is 0.03. How many bad eggs are expected in 500 eggs?
    7. Which of these numbers cannot be a probability?
    8. In a Venn diagram, n(ξ) = 30, n(A) = 12, n(B) = 9 and n(A ∩ B) = 4. What is P(A ∪ B)?
    9. How can you get a better estimate of a probability from an experiment?
    10. The probability of picking a green counter is 0.2. There are 12 green counters. How many counters in total?
    11. A group of 50 students has n(A) = 20 and n(A ∩ B) = 8. What is P(A ∩ B)?
    12. A fair dice is rolled 60 times. About how many sixes should you expect?

    Source: Cambridge IGCSE Mathematics Core and Extended (4th edition, Pimentel and Wall), Chapter 33 (Probability). Explanations, the Try it answers, the tool and the quiz are Guru Jazzy's own. This page is not made or endorsed by Cambridge or Hodder Education.