Moderate -0.3 This is a straightforward application of binomial probability to find a sampling distribution. Students need to recognize that range can only be 0 (all same) or 1 (mix of 1s and 2s), then calculate P(all 1s) = 0.65³ and P(all 2s) = 0.35³ for range 0. The calculations are simple and the conceptual demand is modest—slightly easier than average S2 material.
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.
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
# Question 6:
| Answer | Mark | Guidance |
|--------|------|----------|
| 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 only | B1 | |
| $P(\text{range}=0) = (0.65)^3 + (0.35)^3 = 0.3175$ or $\dfrac{127}{400}$; either range | M1, A1cao | 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. $\dfrac{254}{800}$ |
| $P(\text{range}=1) = (0.35)^2(0.65)\times 3 + (0.65)^2(0.35)\times 3 = 0.6825$ or $\dfrac{273}{400}$ | A1cao | 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 |
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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.\\
\hfill \mbox{\textit{Edexcel S2 2012 Q6 [6]}}