S2 January 2010 Q5

EdexcelOld spec9 marksPoisson Distribution

5. A café serves breakfast every morning. Customers arrive for breakfast at random at a rate of 1 every 6 minutes.

Find the probability that

(a) fewer than 9 customers arrive for breakfast on a Monday morning between 10 am and 11 am. (3)

The café serves breakfast every day between 8 am and 12 noon.

(b) Using a suitable approximation, estimate the probability that more than 50 customers arrive for breakfast next Tuesday. (6)

Mark scheme (a)

Scheme	Marks
\(X \sim \mathrm{Po}(10)\)	B1
\(\mathrm{P}(X \lt 9) = \mathrm{P}(X \leqslant 8)\)	M1
\(= 0.3328\)	A1
	(3)

Notes

B1 for using Po(10)
M1 for attempting to find \(\mathrm{P}(X \leqslant 8)\) : useful values \(\mathrm{P}(X \leqslant 9)\) is 0.4579(M0), using Po(6) gives 0.8472, (M1).
A1 awrt 0.333 but do not accept \(\dfrac{1}{3}\)

Mark scheme (b)

Scheme	Marks
\(Y \sim \mathrm{Po}(40)\) \(Y\) is approximately N(40,40)	M1 A1
\(\mathrm{P}(Y \gt 50) = 1 - \mathrm{P}(Y \leqslant 50)\) \(= 1 - \mathrm{P}\left(Z \lt \dfrac{50.5 - 40}{\sqrt{40}}\right)\)	M1 M1 A1
\(= 1 - \mathrm{P}(Z \lt 1.660..)\) \(= 1 - 0.9515\) \(= 0.0485\)	A1
	(6)
	(9 marks)

Notes

N.B. Calculator gives 0.048437.
Poisson gives 0.0526 (but scores nothing)

1st M1 for identifying the normal approximation
1st A1 for [mean = 40] and [sd = \(\sqrt{40}\) or var = 40]
NB These two marks are B1 M1 on ePEN

These first two marks may be given if the following are seen in the standardisation formula : 40 and \(\sqrt{40}\) or awrt 6.32

2nd M1 for attempting a continuity correction (50 or 30 \(\pm\) 0.5 is acceptable)
3rd M1 for standardising using their mean and their standard deviation and using either 49.5, 50 or 50.5. (29.5, 30, 30.5) accept \(\pm\)

2nd A1 correct z value awrt \(\pm\)1.66 or this may be awarded if see \(\pm\dfrac{50.5 - 40}{\sqrt{40}}\) or \(\pm\dfrac{29.5 - 40}{\sqrt{40}}\)

3rd A1 awrt 3 sig fig in range 0.0484 – 0.0485