S2 January 2010 Q7

EdexcelOld spec11 marksDRVs

7. A bag contains a large number of coins. It contains only 1p and 2p coins in the ratio 1:3

(a) Find the mean \(\mu\) and the variance \(\sigma^2\) of the values of this population of coins. (3)

A random sample of size 3 is taken from the bag.

(b) List all the possible samples. (2)

Mark scheme (a)

Scheme

Marks

\(x\)	1p	2p
\(\mathrm{P}(X = x)\)	\(\dfrac{1}{4}\)	\(\dfrac{3}{4}\)

\(\mu = 1 \times \dfrac{1}{4} + 2 \times \dfrac{3}{4} = \dfrac{7}{4}\) or \(1\frac{3}{4}\) or 1.75

\(\sigma^2 = 1^2 \times \dfrac{1}{4} + 2^2 \times \dfrac{3}{4} - \left(\dfrac{7}{4}\right)^2\)

\(= \dfrac{3}{16}\) or 0.1875

(3)

Notes

B1 1.75 oe
M1 for using \(\sum\left(x^2 p\right) - \mu^2\)
A1 0.1875 oe

Mark scheme (b)

Scheme	Marks
(1,1,1), (1,1,2) any order, (1,2,2) any order, (2,2,2)	B1
(1,2,1) (2,1,1) (2,1,2) (2,2,1) all 8 cases considered. May be implied by 3 * (1,1,2) and 3*(1,2,2)	B1
	(2)

Notes

ignore repeats

Mark scheme (c)

Scheme

Marks

\(\bar{x}\)	1	\(\dfrac{4}{3}\)	\(\dfrac{5}{3}\)	2
\(\mathrm{P}(\bar{X} = \bar{x})\)	\(\frac{1}{4} \times \frac{1}{4} \times \frac{1}{4} = \frac{1}{64}\)	\(3 \times \frac{1}{4} \times \frac{1}{4} \times \frac{3}{4} = \frac{9}{64}\)	\(3 \times \frac{1}{4} \times \frac{3}{4} \times \frac{3}{4} = \frac{27}{64}\)	\(\frac{3}{4} \times \frac{3}{4} \times \frac{3}{4} = \frac{27}{64}\)

B1
M1 A1
M1 A1A1

(6)

(11 marks)

Notes

1st B1 4 correct means (allow repeats)
1st M1 for \(p^3\) for either of the ends
1st A1 for 1/64or awrt 0.016 and 27/64 or awrt 0.422
2nd M1 \(3 \times p^2(1 - p)\) for either of the middle two \(0 \lt p \lt 1\)
May be awarded for finding the probability of the 3 samples with mean of either 4/3 or 5/3.
2nd A1 for 9/64 (or 3/64 three times) and 27/64 (or 9/64 three times) accept awrt 3dp.
3rd A1 fully correct table, accept awrt 3dp.