S2 June 2007 Q4
4. A bag contains a large number of coins:
75% are 10p coins,
25% are 5p coins.
A random sample of 3 coins is drawn from the bag.
Find the sampling distribution for the median of the values of the 3 selected coins. (7)
| Scheme | Marks |
|---|---|
| Attempt to write down combinations | M1 |
| (5,5,5), (5,5,10) any order (10,10,5) any order, (10,10,10) | A1 |
| (5,10,5), (10,5,5), (10,5,10), (5,10,10), | A1 |
| median 5 and 10 | B1 |
| Median = 5 \(\mathrm{P}(M = m) = \left(\dfrac{1}{4}\right)^3 + 3\left(\dfrac{1}{4}\right)^2\left(\dfrac{3}{4}\right) = \dfrac{10}{64} = 0.15625\) | M1 A1 |
| Median = 10 \(\mathrm{P}(M = m) = \left(\dfrac{3}{4}\right)^3 + 3\left(\dfrac{3}{4}\right)^2\left(\dfrac{1}{4}\right) = \dfrac{54}{64} = 0.84375\) | A1 |
| (7) | |
| (7 marks) |
Notes
M1 at least one seen
2nd A1 all 8 cases considered. May be implied by \(3 *\) (10,5,10) and \(3 *\) (5,5,10)
2nd M1 add at least two prob using ¼ and ¾, identified by having same median of 5 or 10. Allow no 3 for M