S2 January 2013 Q1

EdexcelOld spec5 marksPoisson Distribution

(a) Write down the conditions under which the Poisson distribution can be used as an approximation to the binomial distribution. (2)

The probability of any one letter being delivered to the wrong house is 0.01

On a randomly selected day Peter delivers 1000 letters.

(b) Using a Poisson approximation, find the probability that Peter delivers at least 4 letters to the wrong house.
Give your answer to 4 decimal places. (3)

Mark scheme (a)

Scheme	Marks
\(n\) large	B1
\(p\) small	B1
	(2)

Notes

B1 Accept \(n\) (the number of trials) large / high / big / \(n \gt 50\) (accept any number larger than 50)

B1 Accept \(p\) (the probability) small / close to 0 / \(p \lt 0.2\) ( accept any number less than 0.2). Do not accept low.

These must appear in part (a).

Mark scheme (b)

Scheme	Marks
Let \(X\) be the random variable the number of letters delivered to the wrong house
\(X \sim \mathrm{B}(1000, 0.01)\)
Po(10)	B1
\(\mathrm{P}(X \geqslant 4) = 1 - \mathrm{P}(X \leqslant 3)\)	M1
\(= 1 - 0.0103\)
\(= 0.9897\)	A1
	(3)
	(5 marks)

Notes

B1 writing or using Po(10)

M1 using a Poisson (\(\lambda\) need not equal 10) and for writing or using \(1 - \mathrm{P}(X \leqslant 3)\). (Do not accept writing \(1 - \mathrm{P}(X \lt 4)\) unless they have used \(1 - \mathrm{P}(X \leqslant 3)\)).

A1 0.9897 cao must be 4 dp

An awrt 0.990 on its own gains B0M0A0 unless there is evidence that Po(10) is used. In which case it gets B1M1A0

Using B(1000,0.01) gives 0.989927…. and gains B0M0A0