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AS & A Level · Practice questions · The geometric distribution

How long until it happens?

Six original Cambridge-style questions on the geometric distribution, 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

Fail \(r - 1\) times, then succeed.

01
Recall
[2 marks]

A basketball player scores from a free throw with probability 0.7. She keeps taking free throws until she scores. State a suitable distribution for the number of throws she takes, and its mean.

  • \(X \sim \text{Geo}(0.7)\), assuming throws are independent with constant probability of scoring. ✓
  • \(\text{E}(X) = \dfrac{1}{0.7} = 1.43\) (3 s.f.). ✓
02
Calculation
[3 marks]

\(X \sim \text{Geo}(0.15)\). Find (a) \(\text{P}(X = 3)\), (b) \(\text{P}(X \le 4)\).

  • (a) \(0.15 \times 0.85^{2} = 0.108\) (3 s.f.). ✓
  • (b) \(1 - 0.85^{4} = 1 - 0.5220 = 0.478\) (3 s.f.). ✓
03
Calculation
[3 marks]

\(X \sim \text{Geo}(p)\), where \(p > 0.5\), and \(\text{P}(X = 2) = 0.24\). Find \(p\) and \(\text{E}(X)\).

  • \(p(1 - p) = 0.24\), so \(p^{2} - p + 0.24 = 0\), \((p - 0.6)(p - 0.4) = 0\). ✓
  • \(p > 0.5\), so \(p = 0.6\). ✓
  • \(\text{E}(X) = \dfrac{1}{0.6} = \dfrac{5}{3}\). ✓
04
Calculation
[3 marks]

The probability that a randomly chosen person has blood group B is 0.12. People are tested one at a time. Find the least number of people who must be tested for the probability of finding at least one person with group B to be at least 0.95.

  • \(X \sim \text{Geo}(0.12)\); need \(\text{P}(X \le n) = 1 - 0.88^{n} \ge 0.95\), i.e. \(0.88^{n} \le 0.05\). ✓
  • \(0.88^{23} = 0.0529\): too big. \(0.88^{24} = 0.0465\): works. ✓
  • Least \(n = 24\). ✓
05
Analysis
[4 marks]

A fair die is thrown until a six appears. Show that the probability that the number of throws is even is \(\tfrac{5}{11}\).

  • \(\text{P}(X \text{ even}) = \text{P}(2) + \text{P}(4) + \dots = \tfrac{1}{6}\left(\tfrac{5}{6}\right) + \tfrac{1}{6}\left(\tfrac{5}{6}\right)^3 + \dots\) ✓
  • This is a geometric series with first term \(\tfrac{5}{36}\) and common ratio \(\left(\tfrac{5}{6}\right)^2 = \tfrac{25}{36}\). ✓
  • Sum to infinity \(= \dfrac{5/36}{1 - 25/36} = \dfrac{5/36}{11/36} = \dfrac{5}{11}\). ✓
06
Analysis
[5 marks]

Ana and Ben take turns to throw a fair die, Ana first. The first person to throw a six wins.

(a) Find the probability that the game lasts more than 4 throws. (b) Find the probability that Ana wins. (c) Find the expected number of throws in a game.

  • (a) The first 4 throws are not sixes: \(\left(\tfrac{5}{6}\right)^4 = 0.482\) (3 s.f.). ✓
  • (b) Ana wins on throw 1, 3, 5, ...: \(\tfrac{1}{6} + \tfrac{1}{6}\left(\tfrac{5}{6}\right)^2 + \dots = \dfrac{1/6}{1 - 25/36} = \dfrac{6}{11}\). ✓
  • (c) The number of throws \(X \sim \text{Geo}\left(\tfrac{1}{6}\right)\), so \(\text{E}(X) = 6\). ✓

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.