| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2017 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Discrete Random Variables |
| Type | Sampling distribution of range or maximum |
| Difficulty | Standard +0.8 This S2 question requires systematic enumeration of sample outcomes and calculation of probabilities for the maximum of three independent discrete random variables. While the probability calculations are straightforward multiplication, identifying all relevant samples (especially for M=4) requires careful case analysis and organization. The conceptual leap to work with the maximum statistic rather than individual values elevates this above routine exercises, though it remains accessible with methodical work. |
| Spec | 5.02a Discrete probability distributions: general5.02b Expectation and variance: discrete random variables |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \((3, 3, 3)\), \((3, 3, 4) \times 3\), \((3, 4, 4) \times 3\), \((4, 4, 4)\) | B1, B1 | 1st B1 for at least 4 correct samples listed e.g. \((3, 3, 3)\) and \((3, 3, 4) \times 3\). 2nd B1 for all 8 correct samples listed (with no extra or incorrect ones given) |
| (b) \(P(M = 3) = (0.5)^3 = 0.125\) | B1 | B1 for \(P(M = 3) = 0.125\) oe Condone e.g. \(X = 3\) and 12.5% |
| \(P(M = 5) = 1 - (0.8)^3\) or \(P(M = 5) = 3(0.2)(0.8)^2 + 3(0.2)^2(0.8) + (0.2)^3\) OR \(P(M = 4) = 3(0.5)^3(0.3) + 3(0.5)(0.3)^2 + (0.3)^3\) | M1 | 1st M1 for a correct expression or a correct probability for \(P(M = 5)\) or \(P(M = 4)\) |
| \(P(M = 4) = 1 - [(P(M = 3) + P(M = 5)]\) or \(P(M = 5) = 1 - [(P(M = 3) + P(M = 4)]\) | M1 | 2nd M1 for a correct expression for third probability of 3, 4 or 5 or if B1 or 1st M1 are scored then award this mark for using the sum of the probs = 1 |
| A1 | A1 for both \(P(M = 4) = 0.387oe\) and \(P(M = 5) = 0.488oe\) |
| Answer | Marks | Guidance |
|---|---|---|
| \(m\) | 3 | 4 |
| \(P(M = m)\) | 0.125 (\(\frac{1}{8}\)) | 0.387 (\(\frac{387}{1000}\)) |
| (c) Mode of \(S_1 = 3\) and Mode of \(M = 5\) | B1 | B1 for both correct modes with clear \(S\) and \(M\) labels |
**(a)** $(3, 3, 3)$, $(3, 3, 4) \times 3$, $(3, 4, 4) \times 3$, $(4, 4, 4)$ | B1, B1 | 1st B1 for at least 4 correct samples listed e.g. $(3, 3, 3)$ and $(3, 3, 4) \times 3$. 2nd B1 for all 8 correct samples listed (with no extra or incorrect ones given)
**(b)** $P(M = 3) = (0.5)^3 = 0.125$ | B1 | B1 for $P(M = 3) = 0.125$ oe Condone e.g. $X = 3$ and 12.5%
| $P(M = 5) = 1 - (0.8)^3$ or $P(M = 5) = 3(0.2)(0.8)^2 + 3(0.2)^2(0.8) + (0.2)^3$ OR $P(M = 4) = 3(0.5)^3(0.3) + 3(0.5)(0.3)^2 + (0.3)^3$ | M1 | 1st M1 for a correct expression or a correct probability for $P(M = 5)$ or $P(M = 4)$
| $P(M = 4) = 1 - [(P(M = 3) + P(M = 5)]$ or $P(M = 5) = 1 - [(P(M = 3) + P(M = 4)]$ | M1 | 2nd M1 for a correct expression for third probability of 3, 4 or 5 or if B1 or 1st M1 are scored then award this mark for using the sum of the probs = 1
| | A1 | A1 for both $P(M = 4) = 0.387oe$ and $P(M = 5) = 0.488oe$
| | |
| $m$ | 3 | 4 | 5 |
| $P(M = m)$ | 0.125 ($\frac{1}{8}$) | 0.387 ($\frac{387}{1000}$) | 0.488 ($\frac{61}{125}$) |
**(c)** Mode of $S_1 = 3$ and Mode of $M = 5$ | B1 | B1 for both correct modes with clear $S$ and $M$ labels
**Total: 7**
---6. At a men's tennis tournament there are 3 , 4 or 5 sets in a match.
Over many years, data collected show that 50\% of matches last for exactly 3 sets, 30\% of matches last for exactly 4 sets and 20\% of matches last for exactly 5 sets.
A random sample of 3 tennis matches is taken. The number of sets in each match is recorded as $S _ { 1 } , S _ { 2 }$ and $S _ { 3 }$ respectively.
The random variable $M$ represents the maximum value of $S _ { 1 } , S _ { 2 }$ and $S _ { 3 }$
\begin{enumerate}[label=(\alph*)]
\item List all the samples where $M \neq 5$
\item Find the sampling distribution of $M$
\item Write down the mode of $S _ { 1 }$ and the mode of $M$
\end{enumerate}
\hfill \mbox{\textit{Edexcel S2 2017 Q6 [7]}}