A2 October 2021 Q4

EdexcelCurrent spec10 marksDRVs Poisson Distribution

4. Members of a photographic group may enter a maximum of 5 photographs into a members only competition.
Past experience has shown that the number of photographs, \(N\), entered by a member follows the probability distribution shown below.

\(n\)	0	1	2	3	4	5
\(\mathrm{P}(N = n)\)	\(a\)	0.2	0.05	0.25	\(b\)	\(c\)

Given that \(\mathrm{E}(4N + 2) = 14.8\) and \(\mathrm{P}(N = 5 \mid N \gt 2) = \dfrac{1}{2}\)

(a) show that \(\mathrm{Var}(N) = 2.76\) (6)

The group decided to charge a 50p entry fee for the first photograph entered and then 20p for each extra photograph entered into the competition up to a maximum of £1 per person. Thus a member who enters 3 photographs pays 90p and a member who enters 4 or 5 photographs just pays £1

Assuming that the probability distribution for the number of photographs entered by a member is unchanged,

(b) calculate the expected entry fee per member. (3)

Bai suggests that, as the mean and variance are close, a Poisson distribution could be used to model the number of photographs entered by a member next year.

Mark scheme (a)

Scheme	Marks	AO
\(4\mathrm{E}(N) + 2 = 14.8\) or \(\mathrm{E}(N) = 3.2\)	M1	3.1a
\(0.2 + 0.1 + 0.75 + 4b + 5c = 3.2\)	M1	1.1b
\(\dfrac{c}{0.25 + b + c} = 0.5\) or \(0.25 = c - b\)	M1	3.1a
\(b = 0.1\) and \(c = 0.35\)
\(\mathrm{E}(N^2) = 1 \times 0.2 + 4 \times 0.05 + 9 \times 0.25 + 16 \times \text{“}0.1\text{”} + 25 \times \text{“}0.35\text{”}\ [= 13]\)	M1	1.1b
\(\mathrm{Var}(N) = \text{“}13\text{”} - \text{“}3.2\text{”}^2\)	dM1	1.1b
\(= 2.76\)*	A1*	2.1
	(6)

Notes

M1: For using the given information to find \(\mathrm{E}(N)\)
ALT \(a + b + c = 0.5\) oe

M1: For use of \(\sum n\mathrm{P}(N = n) = \text{“}3.2\text{”}\) At least 3 terms correct
ALT \(\sum(4n + 2)\mathrm{P}(N = n) = 14.8 \Rightarrow 2a + 1.2 + 0.5 + 3.5 + 18b + 22c = 14.8\) At least 3 terms correct

M1: Forming an equation in \(b\) and \(c\) using conditional probability

M1: For using \(\sum n^2\mathrm{P}(N = n)\) Allow with the letters \(b\) and \(c\)

dM1: Dependent on previous method mark. Correct method to find \(\mathrm{Var}(N)\)

A1*: All previous marks must be awarded and 2.76 stated

Mark scheme (b)

Scheme

Marks

fee	0	50	70	90	100	100
\(\mathrm{P}(N = n)\)	\(a\)	0.2	0.05	0.25	\(b\)	\(c\)

3.3

\(50 \times 0.2 + 70 \times 0.05 + 90 \times 0.25 + 100 \times \text{“}0.1\text{”} + 100 \times \text{“}0.35\text{”}\)

1.1b

= 81p

1.1b

(3)

Notes

M1: Setting up a new model with the correct fees. At least 3 terms correct. Allow 0.5, 0.7, 0.9, 1

M1: Correct method for calculating E(fee) Allow with the letters \(b\) and \(c\)

A1: 81[p] No units needed. Allow 0.81 if fees are in pounds

Scheme	Marks	AO
Poisson distribution will assign substantial probability to \(N \gt 5\)	B1	3.5b
	(1)
	(10 marks)