IGCSE · Practice questions · Conditional probability

Shrink the group, then count.

Extended. Six original Cambridge-style questions on conditional probability from Venn diagrams, tables, sample spaces and tree diagrams.

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

The given group is the new total.

Spot the condition: 'given that', 'if', 'of those who'.
Restrict: keep only the outcomes in the given group.
Divide: number in both \(\div\) number in the given group.
Trees: second branches are already conditional.
Reverse a tree: wanted path \(\div\) sum of all paths giving the condition.

Recall

[2 marks]

In a Venn diagram of 30 students, 6 do chess only, 4 do chess and art, 11 do art only and 9 do neither.

Find the probability that a student does art, given that they do chess.

Chess group: \(6 + 4 = 10\) students. ✓
4 of them do art: \(\tfrac{4}{10} = \tfrac{2}{5}\) ✓

Calculation

[3 marks]

200 people were asked which phone brand they use.

	Brand A	Brand B
Under 30	48	32
30 or over	54	66

Find (a) P(brand A, given under 30) (b) P(30 or over, given brand B).

(a) Under 30: 80 people, 48 use A: \(\tfrac{48}{80} = \tfrac{3}{5}\) ✓
(b) Brand B: \(32 + 66 = 98\) people, 66 are 30 or over: \(\tfrac{66}{98} = \tfrac{33}{49}\) ✓

Calculation

[2 marks]

A bag holds 6 red and 4 blue counters. Two are taken without replacement.

(a) Given that the first counter is blue, find the probability that the second is blue. (b) Find the probability that both are blue.

(a) 9 left, 3 blue: \(\tfrac{3}{9} = \tfrac{1}{3}\) ✓
(b) \(\tfrac{4}{10} \times \tfrac{1}{3} = \tfrac{4}{30} = \tfrac{2}{15}\) ✓

Calculation

[3 marks]

Two fair dice are rolled. Given that at least one of the dice shows a 6, find the probability that the total is 10 or more.

Cells with at least one 6: \(6 + 6 - 1 = 11\). ✓
Totals of 10 or more among them: (4,6), (6,4), (5,6), (6,5), (6,6), so 5. ✓
Probability \(\tfrac{5}{11}\) ✓

Analysis

[4 marks]

Machine A makes 60% of the items in a factory and machine B makes the rest. 3% of the items from A are faulty and 5% of the items from B are faulty.

(a) Find the probability that an item chosen at random is faulty. (b) Given that an item is faulty, find the probability that it was made by machine B.

Paths: A and faulty \(0.6 \times 0.03 = 0.018\); B and faulty \(0.4 \times 0.05 = 0.02\). ✓
(a) \(0.018 + 0.02 = 0.038\) ✓
(b) \(\dfrac{0.02}{0.038} = 0.526\) (3 s.f.) ✓

Analysis

[4 marks]

In a school, the probability that a student studies History is 0.5, Geography is 0.4, and both is 0.15.

Find the probability that a student (a) studies Geography, given that they study History (b) studies History, given that they do NOT study Geography.

Venn regions: History only 0.35, both 0.15, Geography only 0.25, neither 0.25. ✓
(a) \(\dfrac{0.15}{0.5} = 0.3\) ✓
(b) Not Geography: \(1 - 0.4 = 0.6\); of this, History only is 0.35. ✓
\(\dfrac{0.35}{0.6} = \dfrac{7}{12}\) (0.583) ✓

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.