S2 January 2005 Q5

EdexcelOld spec13 marksPoisson Distribution

5. From company records, a manager knows that the probability that a defective article is produced by a particular production line is 0.032.

A random sample of 10 articles is selected from the production line.

(a) Find the probability that exactly 2 of them are defective. (3)

On another occasion, a random sample of 100 articles is taken.

(b) Using a suitable approximation, find the probability that fewer than 4 of them are defective. (4)

At a later date, a random sample of 1000 is taken.

Mark scheme (a)

Scheme	Marks
Let \(X\) represent the number of defective articles \(\therefore X \sim \mathrm{B}(10, 0.032)\)
\(\mathrm{P}(X = 2) = {}^{10}\mathrm{C}_2(0.032)^2(1 - 0.032)^8\)	M1 A1
\(= \underline{0.0355274\ldots}\)	A1
	(3)

M1 use of \({}^n\mathrm{C}_r\,p^r q^{n-r}\)

1st A1 all correct

2nd A1 awrt 0.0355

Mark scheme (b)

Scheme	Marks
Large \(n\), small \(p \Rightarrow\) Poisson approximation with \(\lambda = 100 \times 0.032 = 3.2\)	B1
\(\mathrm{P}(X \lt 4) = \mathrm{P}(X \leqslant 3) = \mathrm{P}(0) + \mathrm{P}(1) + \mathrm{P}(2) + \mathrm{P}(3)\)	M1
\(= \mathrm{e}^{-3.2}\left\{1 + 3.2 + \dfrac{(3.2)^2}{2} + \dfrac{(3.2)^3}{6}\right\}\)	A1
\(= \underline{0.602519\ldots}\)	A1
	(4)

B1 seen or implied

M1 \(\mathrm{P}(X \leqslant 3)\) stated or implied

1st A1 all correct

2nd A1 awrt 0.603

NB Normal Approx \(\Rightarrow\) 0/4

Mark scheme (c)

Scheme	Marks
\(np\) & \(nq\) both \(\gt 5 \Rightarrow\) Normal approximation with \(np = 32\) and \(npq = 30.976\)	M1 A1
\(\mathrm{P}(X \gt 42) \approx \mathrm{P}(Y \gt 42.5)\) where \(Y \sim \mathrm{N}(32, 30.976)\) \(= \mathrm{P}\left(Z \gt \dfrac{42.5 - 32}{\sqrt{30.976}}\right)\)	M1 A1
\(= \mathrm{P}(Z \gt 1.8865\ldots)\)	A1
\(= \underline{0.0294}\)	A1
	(6)
	(13 marks)

1st M1 N approx

1st A1 both

2nd M1 standardisation, their \(np\), \(\sqrt{npq}\)

2nd A1 all correct

3rd A1 awrt 1.89

4th A1 0.0294–0.0297