IGCSE · Practice questions · Combined events: sample spaces and Venn diagrams

Every outcome counted.

Six original Cambridge-style questions on combined events using sample space diagrams and Venn diagrams with two sets.

Original questions All questions on this page are original work, written in the Cambridge IGCSE style. They are not from past papers. They test the same concepts and skills the syllabus rewards.

The method

Draw it, count it, divide.

Sample space: a grid of every pair of outcomes; each cell equally likely.
Order matters: (2, 6) and (6, 2) are separate cells.
Venn: overlap first, then each circle minus the overlap, then the outside.
Check: all regions add up to the total.
Notation (Extended): \(\cap\) and, \(\cup\) or, \('\) not.

Recall

[2 marks]

A fair coin is thrown and a fair spinner numbered 1, 2, 3 is spun.

(a) List all the possible outcomes. (b) Find the probability of a tail and a 3.

(a) H1, H2, H3, T1, T2, T3 ✓
(b) One of 6 equally likely outcomes: \(\tfrac{1}{6}\) ✓

Calculation

[2 marks]

Two fair dice are rolled and the scores added. Find the probability that the total is (a) 7 (b) a prime number.

(a) (1,6), (2,5), (3,4), (4,3), (5,2), (6,1): \(\tfrac{6}{36} = \tfrac{1}{6}\) ✓
(b) Totals 2, 3, 5, 7, 11 occur \(1 + 2 + 4 + 6 + 2 = 15\) times: \(\tfrac{15}{36} = \tfrac{5}{12}\) ✓

Calculation

[3 marks]

Of 50 students, 28 are in the maths club, 19 are in the science club and 11 are in both.

A student is chosen at random. Find the probability that the student is in (a) neither club (b) exactly one club.

Maths only \(28 - 11 = 17\), science only \(19 - 11 = 8\), both 11. ✓
Neither \(= 50 - (17 + 11 + 8) = 14\). ✓
(a) \(\tfrac{14}{50} = \tfrac{7}{25}\) ✓
(b) \(\tfrac{17 + 8}{50} = \tfrac{25}{50} = \tfrac{1}{2}\) ✓

Analysis

[3 marks]

In a group of 32 students, 20 swim, 15 cycle and 4 do neither.

Find the probability that a student chosen at random swims but does not cycle.

Let \(x\) do both: \((20 - x) + x + (15 - x) + 4 = 32\), so \(39 - x = 32\) and \(x = 7\). ✓
Swim only \(= 20 - 7 = 13\). ✓
Probability \(= \tfrac{13}{32}\) ✓

Calculation

[3 marks]

(Extended) \(\text{P}(A) = 0.5\), \(\text{P}(B) = 0.4\) and \(\text{P}(A \cap B) = 0.15\). Find (a) \(\text{P}(A \cup B)\) (b) \(\text{P}(A \cup B)'\) (c) \(\text{P}(A' \cap B)\).

A only \(= 0.35\), B only \(= 0.25\), both 0.15. ✓
(a) \(0.35 + 0.15 + 0.25 = 0.75\) ✓
(b) \(1 - 0.75 = 0.25\) ✓
(c) B but not A: \(0.25\) ✓

Analysis

[4 marks]

In a game, two fair dice are rolled and the score is the larger of the two numbers (if they are equal, the score is that number).

(a) Find the probability that the score is 6. (b) Find the probability that the score is 3 or less. (c) The game is played 180 times. How many scores of 6 would you expect?

(a) Score 6 when at least one dice shows 6: the last row and column of the grid, \(6 + 6 - 1 = 11\) cells, so \(\tfrac{11}{36}\). ✓
(b) Both dice 3 or less: \(3 \times 3 = 9\) cells, so \(\tfrac{9}{36} = \tfrac{1}{4}\). ✓
(c) \(\tfrac{11}{36} \times 180 = 55\) ✓

Mark this once you have attempted all six and checked your working. It records a Practiced badge on the topic and adds a one-time bonus. Revealing the solutions alone does not count.