S2 June 2012 Q6
6. A bag contains a large number of balls.
65% are numbered 1
35% are numbered 2
A random sample of 3 balls is taken from the bag.
Find the sampling distribution for the range of the numbers on the 3 selected balls. (6)
| Scheme | Marks |
|---|---|
| Attempt to write down combinations at least one seen | M1 |
| (1,1,1), (1,1,2) any order (1,2,2) any order, (2,2,2) no extra combinations | A1 |
| Range 0 and 1 0 and 1 only | B1 |
| [P(range = 0) =] \((0.65)^3 + (0.35)^3\) either range | M1 |
| \(= 0.3175\) or \(\dfrac{127}{400}\) | A1cao |
| [P(range = 1) =] \((0.35)^2(0.65) \times 3 + (0.65)^2(0.35) \times 3\) \(= 0.6825\) or \(\dfrac{273}{400}\) | A1cao |
| (6) | |
| (6 marks) |
Notes
First M1 may be implied by either \((0.65)^3\) or \((0.35)^3\) or \((0.65)^2(0.35)\) or \((0.35)^2(0.65)\)
First A1 may be implied by \((0.65)^3\) and \((0.35)^3\) and \((0.65)^2(0.35)\) and \((0.35)^2(0.65)\)
No need for \(\times 3\)
2nd M1 \((p)^3 + (1 - p)^3\) or \((1 - p)^2(p) \times 3 + (p)^2(1 - p) \times 3\)
A1 for 0.3175 cao or exact equivalent e.g \(\dfrac{254}{800}\)
A1 for 0.6825 cao or exact equivalent e.g \(\dfrac{546}{800}\)
NB These probabilities do not need to be associated with the correct range