S2 June 2016 Q6

EdexcelOld spec10 marksBinomial Distribution DRVs

6. A bag contains a large number of counters with one of the numbers 4, 6 or 8 written on each of them in the ratio 5 : 3 : 2 respectively.

A random sample of 2 counters is taken from the bag.

(a) List all the possible samples of size 2 that can be taken. (2)

The random variable \(M\) represents the mean value of the 2 counters.
Given that \(\mathrm{P}(M = 4) = \dfrac{1}{4}\) and \(\mathrm{P}(M = 8) = \dfrac{1}{25}\)

(b) find the sampling distribution for \(M\). (5)

A sample of \(n\) sets of 2 counters is taken. The random variable \(Y\) represents the number of these \(n\) sets that have a mean of 8

Mark scheme (a)

Scheme	Marks
44, 46, 48, 66, 68, 88 NB 64 is the same as 46, 84 is the same as 48, 86 is the same as 68	B1B1
	(2)

Notes

B1: At least 4 different pairs (ignore incorrect extras)

B1: 6 different pairs with no incorrect extras

Mark scheme (b)

Scheme

Marks

\(\bar{x}\)	4	5	6	7	8
		\(\dfrac{1}{2}\times\dfrac{3}{10}\times2\)	\(\dfrac{3}{10}\times\dfrac{3}{10} + \dfrac{1}{2}\times\dfrac{1}{5}\times2\)	\(\dfrac{3}{10}\times\dfrac{1}{5}\times2\)
\(\mathrm{P}(\bar{X} = \bar{x})\)	\(\dfrac{1}{4}\)	\(\dfrac{3}{10}\)	\(\dfrac{29}{100}\)	\(\dfrac{3}{25}\)	\(\dfrac{1}{25}\)

B1
B1
M1
M1A1

(5)

Notes

B1: 4,5,6,7,8 only no extras or omissions

B1: Writing or using \(\mathrm{P}(X = 4) = \dfrac{1}{2}\), \(\mathrm{P}(X = 6) = \dfrac{3}{10}\) and \(\mathrm{P}(X = 8) = \dfrac{1}{5}\) May be seen in(a)

M1: A correct method for one of P(5), P(6) or P(7) may be implied by correct answer

M1: A correct method for two of P(5), P(6) or P(7) may be implied by correct answer

A1: fully correct table/list -need 4,5,6,7, 8 and their associated probabilities

Mark scheme (c)

Scheme	Marks
\(1 - \left(\dfrac{24}{25}\right)^n \gt 0.9\) or \(\left(\dfrac{24}{25}\right)^n \lt 0.1\) oe	M1
\(n \gt 56.4\)	A1
\(n = 57\)	A1
	(3)
	(10 marks)

Notes

M1: \(1 - \left(\dfrac{24}{25}\right)^n \gt 0.9\) or \(\left(\dfrac{24}{25}\right)^n \lt 0.1\) oe seen or used may use = or \(\leqslant\) instead of <, = or \(\geqslant\) instead of >. Do Not award \(\left(\dfrac{24}{25}\right)^n \gt 0.1\) oe

A1: Ignore any \(n \gt\), \(n \lt\), \(n =\) etc. Award if you see awrt 56.4 may be implied by \(n = 57\)

A1: cao \(n = 57\) or 57 on its own. Do not allow \(n \gt 57\) or \(n \lt 57\). Do not award if alternative values are given. You must check there is no incorrect working

Alternative – trial and error

Scheme

Marks

50	0.87	0.13
51	0.875	0.125
52	0.88	0.12
53	0.885	0.115
54	0.89	0.11
55	0.894	0.106
56	0.898	0.102
57	0.902	0.098
58	0.906	0.094
59	0.91	0.09
60	0.914	0.086

Allow awrt

M1
A1

\(n = 57\)

M1 at least 2 trials for \(50 \leqslant n \leqslant 60\) shown with correct probabilities

A1 trial for \(n = 56\) and 57 shown with correct probabilities

A1: cao \(n = 57\) or 57 on its own. Do not allow \(n \gt 57\) or \(n \lt 57\). Do not award if alternative values are given

(corrected from the printed mark scheme: the table prints 0.865 for \(n = 51\) and 0.94 for \(n = 60\); \(1 - 0.96^{51} = 0.875\) and \(1 - 0.96^{60} = 0.914\))