S2 June 2012 Q4

EdexcelOld spec14 marksPoisson Distribution

4. The number of houses sold by an estate agent follows a Poisson distribution, with a mean of 2 per week.

(a) Find the probability that in the next 4 weeks the estate agent sells,

(i) exactly 3 houses,

(ii) more than 5 houses.

(5)

The estate agent monitors sales in periods of 4 weeks.

(b) Find the probability that in the next twelve of these 4 week periods there are exactly nine periods in which more than 5 houses are sold. (3)

The estate agent will receive a bonus if he sells more than 25 houses in the next 10 weeks.

Mark scheme (a)

Scheme	Marks
Let \(X\) be the random variable the number of houses sold.
\(X \sim \mathrm{Po}(8)\)	B1
(i) \(\mathrm{P}(X \leqslant 3) - \mathrm{P}(X \leqslant 2) = 0.0424 - 0.0138\) or \(\dfrac{\mathrm{e}^{-8}8^3}{3!}\)	M1
\(= 0.0286\) awrt 0.0286	A1
(ii) \(\mathrm{P}(X \gt 5) = 1 - \mathrm{P}(X \leqslant 5)\)	M1
\(= 1 - 0.1912\)
\(= 0.8088\) awrt 0.809	A1
	(5)

Notes

1st B1 for writing or using Po(8) in either (i) or (ii)

(i) M1 writing or using \(\mathrm{P}(X \leqslant 3) - \mathrm{P}(X \leqslant 2)\) or \(\dfrac{e^{-8}8^3}{3!}\)

(ii) M1 writing or using \(1 - \mathrm{P}(X \leqslant 5)\)

Mark scheme (b)

Scheme	Marks
Let \(Y\) be the random variable = the number of periods where more than 5 houses are sold
\(Y \sim \mathrm{B}(12, 0.8088)\)	M1
\(\mathrm{P}(Y = 9) = (0.8088)^9(1 - 0.8088)^3\dfrac{12!}{9!3!}\)	M1
\(= 0.228\) awrt 0.228	A1
	(3)

Notes

M1 writing or attempting to use B(12,their (a(ii))) NB ft their a(ii) to at least 2sf

M1 \(\dfrac{12!}{9!3!}\)(a(ii))\(^9\)(1- a(ii))\(^3\) allow \({}^{12}\mathrm{C}_3\) or \({}^{12}\mathrm{C}_9\) or 220 instead of \(\dfrac{12!}{9!3!}\) NB ft their a(ii) to at least 1sf but an expression must be seen (No use of tables)

Mark scheme (c)

Scheme	Marks
N(20,20)	M1A1
\(\mathrm{P}(X \gt 25) = 1 - \mathrm{P}\left(Z \leqslant \dfrac{25.5 - 20}{\sqrt{20}}\right)\)	M1,M1,A1
\(= 1 - \mathrm{P}(Z \leqslant 1.23)\)
\(= 1 - 0.8907\)
\(= 0.1093 / 0.1094\) awrt 0.109	A1
	(6)
	(14 marks)

Notes

1st M1 for writing or using a normal approximation

1st A1 for correct mean and sd (may be given if correct in standardisation formula)

2nd M1 Standardising using their mean and their sd and using [24.5, 25, 25.5, 26 or 26.5] and for finding correct area by doing \(1 - \mathrm{P}(Z \leqslant\) “their 1.23”)

NB if they have not written down a mean and sd then they need to be correct in the standardisation to gain this mark.

3rd M1 for attempting a continuity correction (\(26 \pm 0.5\))

2nd A1 for \(\pm\dfrac{25.5 - 20}{\sqrt{20}}\) or \(\pm\) awrt 1.2 or better.

SC using \(\mathrm{P}(X \lt 26.5/25.5) - \mathrm{P}(X \lt 25.5/24.5)\) can get M1A1 M0M1A0A0