S2 June 2013 (R) Q1
1. A bag contains a large number of counters. A third of the counters have a number 5 on them and the remainder have a number 1.
A random sample of 3 counters is selected.
| Scheme | Marks |
|---|---|
| (1, 1, 1), (5, 5, 5), (1, 5, 5), (1, 5, 1) | B1 |
| (1,1,1); (5,5,5); (1, 5, 5); (5, 1, 5); (5, 5, 1) (5, 1, 1); (1, 5, 1); (1, 1, 5) | B1 |
| (2) |
Notes
1st B1 for any two of the triples
2nd B1 for all 8 cases. No incorrect extras – condone repeats. Allow (1, 5, 5) (x 3) and (1, 1, 5) (x 3) instead of writing all three cases down
| Scheme | Marks |
|---|---|
| \(r\): 0 and 4 | B1 |
| \(\mathrm{P}(R = 0) = \dfrac{9}{27}\) or \(\dfrac{1}{3}\) \(\mathrm{P}(R = 4) = \dfrac{18}{27}\) or \(\dfrac{2}{3}\) | M1d A1 |
| (3) | |
| (5 marks) |
Notes
B1 for both values of \(r\)
M1 d dependent on previous B1. For an attempt to evaluate one of the probabilities for \(r\) correctly e.g. for \(r = 0\); \(\left(\dfrac{2}{3}\right)^3 + \left(\dfrac{1}{3}\right)^3\) and for \(r = 4\); \(3 \times \left(\dfrac{1}{3}\right)^2 \times \left(\dfrac{2}{3}\right) + 3 \times \left(\dfrac{1}{3}\right) \times \left(\dfrac{2}{3}\right)^2\) Working must be shown.
A1 for both values of \(r\) and their correct corresponding probabilities. Allow awrt 0.333 and 0.667
NB Correct answer with no working will gain B1M0A0