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Date	May 2022	Marks available	3	Reference code	22M.1.AHL.TZ1.12
Level	Additional Higher Level	Paper	Paper 1	Time zone	Time zone 1
Command term	Find	Question number	12	Adapted from	N/A

Question

The strength of earthquakes is measured on the Richter magnitude scale, with values typically between $0$ and $8$ where $8$ is the most severe.

The Gutenberg–Richter equation gives the average number of earthquakes per year, $N$ , which have a magnitude of at least $M$ . For a particular region the equation is

$\log_{10} N = a - M$ , for some $a \in ℝ$ .

This region has an average of $100$ earthquakes per year with a magnitude of at least $3$ .

The equation for this region can also be written as $N = \frac{b}{10^{M}}$ .

Within this region the most severe earthquake recorded had a magnitude of $7.2$ .

The number of earthquakes in a given year with a magnitude of at least $7.2$ can be modelled by a Poisson distribution, with mean $N$ . The number of earthquakes in one year is independent of the number of earthquakes in any other year.

Let $Y$ be the number of years between the earthquake of magnitude $7.2$ and the next earthquake of at least this magnitude.

Find the value of $a$ .

[2]

Find the value of $b$ .

[2]

Find the average number of earthquakes in a year with a magnitude of at least $7.2$ .

[1]

Find $P (Y > 100)$ .

[3]

Markscheme

$\log_{10} 100 = a - 3$ (M1)

$a = 5$ A1

[2 marks]

EITHER

$N = 10^{5 - M}$ (M1)

$= \frac{10^{5}}{10^{M}} (= \frac{100000}{10^{M}})$

$100 = \frac{b}{10^{3}}$ (M1)

THEN

$b = 100000 (= 10^{5})$ A1

[2 marks]

$N = \frac{10^{5}}{10^{7.2}} = 0.00631 (0.0063095 \dots)$ A1

Note: Do not accept an answer of $10^{- 2.2}$ .

[1 mark]

METHOD 1

$Y > 100 \Rightarrow$ no earthquakes in the first $100$ years (M1)

EITHER

let $X$ be the number of earthquakes of at least magnitude $7.2$ in a year

$X ~ Po (0.0063095 \dots)$

${(P (X = 0))}^{100}$ (M1)

let $X$ be the number of earthquakes in $100$ years

$X ~ Po (0.0063095 \dots \times 100)$ (M1)

$P (X = 0)$

THEN

$0.532 (0.532082 \dots)$ A1

METHOD 2

$Y > 100 \Rightarrow$ no earthquakes in the first $100$ years (M1)

let $X$ be the number of earthquakes in $100$ years

since $n$ is large and $p$ is small

$X ~ B (100, 0.0063095 \dots)$ (M1)

$P (X = 0)$

$0.531 (0.531019 \dots)$ A1

[3 marks]

Examiners report

Parts (a), (b), and (c) were accessible to many candidates who earned full marks with the manipulation of logs and indices presenting no problems. Part (d), however, proved to be too difficult for most and very few correct attempts were seen. As in question 9, most candidates relied on calculator notation when using the Poisson distribution. The discipline of defining a random variable in terms of its distribution and parameters helps to conceptualize the problem in terms that aid a better understanding. Most candidates who attempted this question blindly entered values into the Poisson distribution calculator and were unable to earn any marks. There were a couple of correct solutions using a binomial distribution to approximate the given quantity.

[N/A]

Syllabus sections

Topic 4—Statistics and probability » AHL 4.17—Poisson distribution

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