S2 January 2012 Q4

EdexcelOld spec16 marksPoisson Distribution

4. A website receives hits at a rate of 300 per hour.

(a) State a distribution that is suitable to model the number of hits obtained during a 1 minute interval. (1)

(b) State two reasons for your answer to part (a). (2)

Find the probability of

(d) at least 15 hits in 2 minutes. (3)

The website will go down if there are more than 70 hits in 10 minutes.

(e) Using a suitable approximation, find the probability that the website will go down in a particular 10 minute interval. (7)

Scheme	Marks
Poisson	B1
	(1)

Mark scheme (b)

Scheme	Marks
Hits occur singly in time Hits are independent or Hits occur randomly Hits occur at a constant rate	B1B1
	(2)

Notes

1st B1 Any one of the 3 statements - no context required. NB It must be a constant (mean) rate and not a constant probability or a constant mean.
2nd B1 A different statement with context of hits. NB random and independent are the same statement.
If only one mark awarded give the 1st B1. Never award B0 B1

Mark scheme (c)

Scheme	Marks
\(X \sim \mathrm{Po}(5)\)	B1
\(P(X = 10) = \mathrm{P}(X \leqslant 10) - \mathrm{P}(X \leqslant 9)\) or \(\dfrac{e^{-5}5^{10}}{10!}\)	M1
\(= 0.9863 - 0.9682\) \(= 0.0181\) awrt 0.0181	A1
	(3)

Notes

B1 writing or using Po(5)
M1 writing or using \(\mathrm{P}(X \leqslant 10) - \mathrm{P}(X \leqslant 9)\) or \(\dfrac{e^{-5}5^{10}}{10!}\)

Mark scheme (d)

Scheme	Marks
\(X \sim \mathrm{Po}(10)\)	B1
\(\mathrm{P}(X \geqslant 15) = 1 - \mathrm{P}(X \leqslant 14)\)	M1
\(= 1 - 0.9165\) \(= 0.0835\) awrt 0 .0835	A1
	(3)

Notes

B1 writing or using Po(10)
M1 writing or using \(1 - \mathrm{P}(X \leqslant 14)\)

Mark scheme (e)

Scheme	Marks
\(X \sim \mathrm{Po}(50)\) Approximated by N(50,50)	B1B1
\(\mathrm{P}(X \gt 70) = \mathrm{P}\left(Z \gt \dfrac{70.5 - 50}{\sqrt{50}}\right)\)	M1M1
\(= \mathrm{P}(Z \gt 2.899...)\)	A1
\(= 1 - 0.9981\)	M1
\(= 0.0019\) awrt 0.0019	A1
	(7)
	(16 marks)

Notes

1st B1 for a normal approximation 2nd B1 for correct mean and sd (may be seen in standardisation formula
1st M1 for attempting a continuity correction (71 \(\pm\) 0.5)
2nd M1 Standardising using their mean and their sd and using [69.5, 70, 70.5, 71 or 71.5] allow \(\pm\) z
NB if they have not written down a mean and sd then they need to be correct in the standardisation to gain this mark.
1st A1 for \(z = \pm\) awrt 2.9 or better. May be awarded for \(\pm\dfrac{70.5 - 50}{\sqrt{50}}\)
3rd M1 for 1 - tables value

SC using P(\(X\)< 70.5/71.5) – P(\(X\)<69.5/70.5) can get B1B1 M0M1A0 M0A0