| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2021 |
| Session | October |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Discrete Random Variables |
| Type | Sampling distribution of mean or linear combination |
| Difficulty | Standard +0.3 This is a straightforward application of discrete probability distributions requiring enumeration of outcomes (9 cases), calculation of probabilities using independence, and finding expectation. The linear combination is simple (4X - 2Y), and all steps follow standard S2 procedures with no novel insight required. Slightly above average difficulty due to the enumeration work and multiple parts, but remains a routine textbook exercise. |
| Spec | 5.01a Permutations and combinations: evaluate probabilities5.03b Solve problems: using pdf5.03c Calculate mean/variance: by integration |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(P(\text{score } 8) = 0.25 \times 0.35 = 0.0875\) | B1 | A correct calculation shown followed by 0.0875 |
| (1) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Correct set of \(y\) values identified | B1 | For identifying correct set of \(y\) values. Any extras must have probability 0. May be split e.g. 2 may appear twice |
| At least two correct calculations or probs from \(P(Y=-2)\), \(P(Y=0)\), \(P(Y=4)\) or \(P(Y=10)\) | M1 | |
| At least one correct calculation or prob for \(P(Y=2)\) or \(P(Y=6)\) | M1 | |
| At least four correct calculations or probs attached to correct \(y\) value or sample | M1 | |
| Full distribution table: \(Y\): \(-2, 0, 2, 4, 6, 8, 10\) with \(P(Y=y)\): \(0.1, 0.14, 0.2475, 0.1225, 0.2025, 0.0875, 0.1\) | A1 | A fully correct answer |
| (5) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(E(Y) = -2 \times 0.1 + 0 \times 0.14 + 2 \times 0.2475 + 4 \times 0.1225 + 6 \times 0.2025 + 8 \times 0.0875 + 10 \times 0.1\) | M1 | Correct expression ft their table. Alt: \(E(X) = 0.4 + 2\times0.35 + 3\times0.25 [=1.85]\) and \(E(Y) = 4\times"1.85" - 2\times"1.85"\) |
| \(= 3.7\) | A1 | 3.7 or exact equivalent |
| (2) | ||
| Total 8 |
# Question 5:
## Part (a)
| Answer | Mark | Guidance |
|--------|------|----------|
| $P(\text{score } 8) = 0.25 \times 0.35 = 0.0875$ | B1 | A correct calculation shown followed by 0.0875 |
| | **(1)** | |
## Part (b)
| Answer | Mark | Guidance |
|--------|------|----------|
| Correct set of $y$ values identified | B1 | For identifying correct set of $y$ values. Any extras must have probability 0. May be split e.g. 2 may appear twice |
| At least two correct calculations or probs from $P(Y=-2)$, $P(Y=0)$, $P(Y=4)$ or $P(Y=10)$ | M1 | |
| At least one correct calculation or prob for $P(Y=2)$ or $P(Y=6)$ | M1 | |
| At least four correct calculations or probs attached to correct $y$ value or sample | M1 | |
| Full distribution table: $Y$: $-2, 0, 2, 4, 6, 8, 10$ with $P(Y=y)$: $0.1, 0.14, 0.2475, 0.1225, 0.2025, 0.0875, 0.1$ | A1 | A fully correct answer |
| | **(5)** | |
## Part (c)
| Answer | Mark | Guidance |
|--------|------|----------|
| $E(Y) = -2 \times 0.1 + 0 \times 0.14 + 2 \times 0.2475 + 4 \times 0.1225 + 6 \times 0.2025 + 8 \times 0.0875 + 10 \times 0.1$ | M1 | Correct expression ft their table. Alt: $E(X) = 0.4 + 2\times0.35 + 3\times0.25 [=1.85]$ and $E(Y) = 4\times"1.85" - 2\times"1.85"$ |
| $= 3.7$ | A1 | 3.7 or exact equivalent |
| | **(2)** | |
| | **Total 8** | |
---\begin{enumerate}
\item A bag contains a large number of counters.
\end{enumerate}
40\% of the counters are numbered 1\\
$35 \%$ of the counters are numbered 2\\
$25 \%$ of the counters are numbered 3
In a game Alif draws two counters at random from the bag. His score is 4 times the number on the first counter minus 2 times the number on the second counter.\\
(a) Show that Alif gets a score of 8 with probability 0.0875\\
(b) Find the sampling distribution of Alif's score.\\
(c) Calculate Alif's expected score.
\hfill \mbox{\textit{Edexcel S2 2021 Q5 [8]}}