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Date May 2021 Marks available 4 Reference code 21M.2.AHL.TZ2.5
Level Additional Higher Level Paper Paper 2 Time zone Time zone 2
Command term Find Question number 5 Adapted from N/A

Question

Hank sets up a bird table in his garden to provide the local birds with some food. Hank notices that a specific bird, a large magpie, visits several times per month and he names him Bill. Hank models the number of times per month that Bill visits his garden as a Poisson distribution with mean 3.1.

Over the course of 3 consecutive months, find the probability that Bill visits the garden:

Using Hank’s model, find the probability that Bill visits the garden on exactly four occasions during one particular month.

[1]
a.

on exactly 12 occasions.

[2]
b.i.

during the first and third month only.

[3]
b.ii.

Find the probability that over a 12-month period, there will be exactly 3 months when Bill does not visit the garden.

[4]
c.

After the first year, a number of baby magpies start to visit Hank’s garden. It may be assumed that each of these baby magpies visits the garden randomly and independently, and that the number of times each baby magpie visits the garden per month is modelled by a Poisson distribution with mean 2.1.

Determine the least number of magpies required, including Bill, in order that the probability of Hank’s garden having at least 30 magpie visits per month is greater than 0.2.

[4]
d.

Markscheme

X1~Po3.1

PX1=4=0.173  0.173349                A1


[1 mark]

a.

X2~Po3×3.1=Po9.3               (M1)

PX2=12=0.0799  0.0798950                A1


[2 marks]

b.i.

PX1>02×PX1=0               (M1)

0.954952×0.04505               (A1)

=0.0411  0.0410817                A1


[3 marks]

b.ii.

PX1=0=0.04505               (A1)

X1~B12, 0.04505               (M1)(A1)


Note:
Award M1 for recognizing binomial probability, and A1 for correct parameters.


=0.0133  0.013283                A1


[4 marks]

c.

METHOD ONE

               (M1)(A1)(A1)


Note: Award M1 for evidence of a cumulative Poisson with λ=3.1+2.1n, A1 for 0.136705 and A1 for 0.253384.


so require 12 magpies (including Bill)               A1

 

METHOD TWO

evidence of a cumulative Poisson with λ=3.1+2.1n               (M1)

sketch of curve and y=0.2               (A1)

(intersect at) 10.5810               (A1)

rounding up gives n=11

so require 12 magpies (including Bill)               A1

 

[4 marks]

d.

Examiners report

[N/A]
a.
[N/A]
b.i.
[N/A]
b.ii.
[N/A]
c.
[N/A]
d.

Syllabus sections

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

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