S2 January 2010 Q1

EdexcelOld spec8 marksBinomial Distribution

1. A manufacturer supplies DVD players to retailers in batches of 20. It has 5% of the players returned because they are faulty.

(a) Write down a suitable model for the distribution of the number of faulty DVD players in a batch. (2)

Find the probability that a batch contains

(b) no faulty DVD players, (2)

(d) Find the mean and variance of the number of faulty DVD players in a batch. (2)

Scheme	Marks
\(X \sim B(20, 0.05)\)	B1 B1
	(2)

Mark scheme (b)

Scheme	Marks
\(\mathrm{P}(X = 0) = 0.95^{20} = 0.3584859\ldots\) or 0.3585 using tables.	M1 A1
	(2)

Notes

M1 for finding \((p)^{20}\) \(0 \lt p \lt 1\) this working needs to be seen if answer incorrect to gain the M1
A1 awrt 0.358 or 0.359.

NB In parts b, c and d correct answers with no working gain full marks

Mark scheme (c)

Scheme	Marks
\(\mathrm{P}(X \gt 4) = 1 - \mathrm{P}(X \leqslant 4)\)	M1
\(= 1 - 0.9974\) \(= 0.0026\)	A1
	(2)

Notes

M1 for writing \(1 - \mathrm{P}(X \leqslant 4)\)
or \(1 - [\,\mathrm{P}(X = 0) + \mathrm{P}(X = 1) + \mathrm{P}(X = 2) + \mathrm{P}(X = 3) + \mathrm{P}(X = 4)]\)
or \(1 - 0.9974\)
or \(1 - 0.9568\)

A1 awrt 0.0026 or \(2.6 \times 10^{-3}\), do not accept a fraction e.g. 26/10000

NB In parts b, c and d correct answers with no working gain full marks

Mark scheme (d)

Scheme	Marks
\(\text{Mean} = 20 \times 0.05 = 1\)	B1
\(\text{Variance} = 20 \times 0.05 \times 0.95 = 0.95\)	B1
	(2)
	(8 marks)

Notes

1st B1 for 1
2nd B1 for 0.95

NB In parts b, c and d correct answers with no working gain full marks