S2 June 2013 Q1
1. A bag contains a large number of 1p, 2p and 5p coins.
50% are 1p coins
20% are 2p coins
30% are 5p coins
A random sample of 3 coins is chosen from the bag.
| Scheme | Marks |
|---|---|
| (5,5,5) or (1,5,5) or (2,5,5) | B1 |
| (5,5,5) (5,5,1) (5,1,5) (1,5,5) (5,5,2) (5,2,5) (2,5,5) or (5,5,5) and (5,5,1) (\(\times 3\)) and (5,5,2) (\(\times 3\)) | B1 |
| (2) |
Notes
1st B1 for two of the given triples, any order
2nd B1 for all 7 cases. no incorrect extras
| Scheme | Marks |
|---|---|
| (5,5,5) \(\left(\tfrac{3}{10}\right)^3 = \dfrac{27}{1000} = 0.027\) | B1 |
| (5,5,1) \(3 \times \tfrac{1}{2} \times \left(\tfrac{3}{10}\right)^2 = \dfrac{135}{1000}\) or \(\dfrac{27}{200} = 0.135\) | M1 |
| (5,5,2) \(3 \times \tfrac{1}{5} \times \left(\tfrac{3}{10}\right)^2 = \dfrac{54}{1000} = \dfrac{27}{500} = 0.054\) | |
| \(\mathrm{P}(M = 5) = \left(\tfrac{3}{10}\right)^3 + 3 \times \tfrac{1}{2} \times \left(\tfrac{3}{10}\right)^2 + 3 \times \tfrac{1}{5} \times \left(\tfrac{3}{10}\right)^2 = \dfrac{27}{125} = 0.216\) oe | A1A1 |
| (4) |
Notes
B1 \(\left(\tfrac{3}{10}\right)^3\) or 0.027 oe. This can be a single term in a summation
M1 either “3” \(\times \tfrac{1}{2} \times \left(\tfrac{3}{10}\right)^2\) or “3” \(\times \tfrac{1}{5} \times \left(\tfrac{3}{10}\right)^2\) oe. May omit the \(3 \times\) or have another positive integer in place of the 3. These may be seen as a single term in a summation
A1 \(\left(\tfrac{3}{10}\right)^3 + 3 \times \tfrac{1}{2} \times \left(\tfrac{3}{10}\right)^2 + 3 \times \tfrac{1}{5} \times \left(\tfrac{3}{10}\right)^2\) oe
A1 0.216 oe
| Scheme | Marks | ||||||||
|---|---|---|---|---|---|---|---|---|---|
| \(\mathrm{P}(M = 1) = (0.5)^3 + 3(0.5)^2(0.2) + 3(0.5)^2(0.3)\) | M1 | ||||||||
| \(= 0.5\) | A1 | ||||||||
| \(\mathrm{P}(M = 2) = \left(\tfrac{1}{5}\right)^3 + 3 \times \left(\tfrac{1}{5}\right)^2 \times \tfrac{1}{2} + 3 \times \left(\tfrac{1}{5}\right)^2 \times \tfrac{3}{10} + 6 \times \tfrac{1}{2} \times \tfrac{1}{5} \times \tfrac{3}{10}\) | M1 | ||||||||
| \(= 0.284\) or \(\dfrac{71}{250}\) oe | A1 | ||||||||
| A1 | ||||||||
| (5) | |||||||||
| (11 marks) |
Notes
1st M1 correct calculation for \(\mathrm{P}(M = 1)\) or \(\mathrm{P}(M = 2)\), working must be shown and not implied by a correct answer.
1st A1 either \(\mathrm{P}(M = 1)\) or \(\mathrm{P}(M = 2)\) correct
2nd M1 correct calculation for both \(\mathrm{P}(M = 1)\) and \(\mathrm{P}(M = 2)\), or their probabilities adding up to 1, but do not allow probabilities of 0.5, 0.2 and 0.3
2nd A1 both \(\mathrm{P}(M = 1)\) and \(\mathrm{P}(M = 2)\) correct
3rd A1dep on both M marks awarded. All three values written down with their correct probabilities. They must be in part (c) but they do not need to be in a table.
NB A fully correct table with no working will get M0 A0 M1 A1 A0.