| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2016 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Discrete Probability Distributions |
| Type | Construct probability distribution from scenario |
| Difficulty | Moderate -0.8 This is a straightforward probability distribution construction requiring systematic enumeration of outcomes from two dice. Part (i) involves counting favorable outcomes for each score (standard S1 technique), and part (ii) is direct application of E(X) = Σxp(x). The scenario is clearly defined with no conceptual subtlety, making it easier than average A-level material. |
| Spec | 2.04a Discrete probability distributions2.04b Binomial distribution: as model B(n,p) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(P(0) = 6/36\), \(P(1) = 10/36\), \(P(2) = 8/36\) | B1 | Table with 0, 1, 2, 3, 4, 5 seen (6 if \(P(6)=0\)) |
| B1 | Any three probs correct | |
| \(P(3) = 6/36\), \(P(4) = 4/36\), \(P(5) = 2/36\) | M1 | \(\sum p = 1\) and at least 3 outcomes |
| A1 [4] | All probs correct |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| mean score \(= \frac{(0\times6 + 1\times10 + 16 + 18 + 16 + 10)}{36}\) | M1 | Using \(\sum xp\) (unsimplified) on its own – condone \(\sum p \neq 1\) |
| \(= 70/36\ (35/18,\ 1.94)\) | A1 [2] |
## Question 3:
### Part (i)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $P(0) = 6/36$, $P(1) = 10/36$, $P(2) = 8/36$ | **B1** | Table with 0, 1, 2, 3, 4, 5 seen (6 if $P(6)=0$) |
| | **B1** | Any three probs correct |
| $P(3) = 6/36$, $P(4) = 4/36$, $P(5) = 2/36$ | **M1** | $\sum p = 1$ and at least 3 outcomes |
| | **A1** [4] | All probs correct |
### Part (ii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| mean score $= \frac{(0\times6 + 1\times10 + 16 + 18 + 16 + 10)}{36}$ | **M1** | Using $\sum xp$ (unsimplified) on its own – condone $\sum p \neq 1$ |
| $= 70/36\ (35/18,\ 1.94)$ | **A1** [2] | |
---3 Two ordinary fair dice are thrown. The resulting score is found as follows.
\begin{itemize}
\item If the two dice show different numbers, the score is the smaller of the two numbers.
\item If the two dice show equal numbers, the score is 0 .\\
(i) Draw up the probability distribution table for the score.\\
(ii) Calculate the expected score.
\end{itemize}
\hfill \mbox{\textit{CAIE S1 2016 Q3 [6]}}