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AS & A Level · Practice questions · Probability: sample spaces, exclusive events and tree diagrams

Branches and counts.

Six original Cambridge-style questions on evaluating probabilities by counting, by the addition and multiplication rules, and with combinations, at Paper 5 standard.

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

Count, then add or multiply.

01
Recall
[2 marks]

Explain the difference between exclusive events and independent events, and state \(\text{P}(A \cap B)\) in each case.

  • Exclusive: \(A\) and \(B\) cannot both happen, so \(\text{P}(A \cap B) = 0\). ✓
  • Independent: the occurrence of one does not affect the probability of the other, so \(\text{P}(A \cap B) = \text{P}(A) \times \text{P}(B)\). ✓
02
Calculation
[4 marks]

Two fair dice are thrown. \(A\) is the event 'the total is 9' and \(B\) is the event 'the scores differ by 2'.

(a) Find \(\text{P}(A)\) and \(\text{P}(B)\). (b) Are \(A\) and \(B\) exclusive? (c) Find \(\text{P}(A \cup B)\).

  • (a) Total 9: (3,6), (4,5), (5,4), (6,3), so \(\text{P}(A) = \tfrac{4}{36} = \tfrac{1}{9}\). ✓
  • Differ by 2: (1,3), (2,4), (3,5), (4,6) and their reverses, so \(\text{P}(B) = \tfrac{8}{36} = \tfrac{2}{9}\). ✓
  • (b) Two scores with total 9 differ by an odd number, so they cannot differ by 2. Yes, exclusive. ✓
  • (c) \(\tfrac{1}{9} + \tfrac{2}{9} = \tfrac{1}{3}\). ✓
03
Calculation
[4 marks]

A box contains 4 red and 6 green counters. Three counters are taken at random without replacement. Find the probability that (a) all three are the same colour, (b) at least one is red.

  • (a) RRR: \(\tfrac{4}{10} \times \tfrac{3}{9} \times \tfrac{2}{8} = \tfrac{24}{720}\). GGG: \(\tfrac{6}{10} \times \tfrac{5}{9} \times \tfrac{4}{8} = \tfrac{120}{720}\). ✓
  • Same colour: \(\tfrac{144}{720} = \tfrac{1}{5}\). ✓
  • (b) \(1 - \text{P}(GGG) = 1 - \tfrac{120}{720} = \tfrac{5}{6}\). ✓
04
Calculation
[3 marks]

Ali and Bo take a driving test on the same day. The probability that Ali passes is 0.7 and the probability that Bo passes is 0.6, independently. Find the probability that (a) exactly one of them passes, (b) at least one of them passes.

  • (a) \(0.7 \times 0.4 + 0.3 \times 0.6 = 0.28 + 0.18 = 0.46\). ✓
  • (b) \(1 - 0.3 \times 0.4 = 1 - 0.12 = 0.88\). ✓
05
Analysis
[4 marks]

Five people are chosen at random from 6 men and 4 women. Find the probability that more women than men are chosen.

  • More women than men means 3 women and 2 men, or 4 women and 1 man. ✓
  • \({}^{4}C_{3}\,{}^{6}C_{2} + {}^{4}C_{4}\,{}^{6}C_{1} = 4 \times 15 + 1 \times 6 = 66\). ✓
  • Total \({}^{10}C_{5} = 252\). Probability \(= \tfrac{66}{252} = \tfrac{11}{42} = 0.262\) (3 s.f.). ✓
06
Analysis
[5 marks]

Mia and Noor play a game. Mia throws a fair die first, then Noor, then Mia, then Noor. The first person to throw a six wins, and the game stops after four throws if nobody has thrown a six.

Find the probability that (a) Mia wins, (b) nobody wins, (c) Noor wins.

  • (a) Mia wins on throw 1: \(\tfrac{1}{6}\). On throw 3: \(\left(\tfrac{5}{6}\right)^2 \tfrac{1}{6} = \tfrac{25}{216}\). ✓
  • \(\tfrac{36}{216} + \tfrac{25}{216} = \tfrac{61}{216} = 0.282\) (3 s.f.). ✓
  • (b) No six in four throws: \(\left(\tfrac{5}{6}\right)^4 = \tfrac{625}{1296} = 0.482\) (3 s.f.). ✓
  • (c) Noor on throw 2: \(\tfrac{5}{36}\); on throw 4: \(\left(\tfrac{5}{6}\right)^3 \tfrac{1}{6} = \tfrac{125}{1296}\). Total \(\tfrac{180 + 125}{1296} = \tfrac{305}{1296} = 0.235\) (3 s.f.). ✓
  • Check: \(\tfrac{366}{1296} + \tfrac{625}{1296} + \tfrac{305}{1296} = 1\). ✓

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.