| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2020 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Discrete Probability Distributions |
| Type | Verify probability from given formula |
| Difficulty | Moderate -0.8 This is a straightforward discrete probability question requiring systematic enumeration of outcomes from two spinners. Part (a) is a guided 'show that' calculation, part (b) involves completing a standard probability distribution table, and part (c) applies standard formulas for expectation and variance. The question requires careful bookkeeping but no novel insight or complex problem-solving—it's easier than average A-level work. |
| Spec | 2.02g Calculate mean and standard deviation2.04a Discrete probability distributions |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Sample space grid shown with max values of paired selections giving entries 1,1,2,2,3 across rows/columns | M1 | |
| \(\frac{7}{15}\) | A1 (AG) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(x\): 1, 2, 3; Probability: \(\frac{2}{15}\), \(\frac{6}{15}\), \(\frac{7}{15}\) | B1 | Table structure |
| \(P(1)\) or \(P(2)\) correct | B1 | |
| 3rd probability correct, FT sum to 1 | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(E(X) = \frac{2+12+21}{15} = \frac{35}{15} = \frac{7}{3}\) | B1 | |
| \(\text{Var}(X) = \frac{1^2 \times 2 + 2^2 \times 6 + 3^2 \times 7}{15} - \left(\frac{7}{3}\right)^2\) | M1 | |
| \(\frac{22}{45}\) \((0.489)\) | A1 |
## Question 5(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| Sample space grid shown with max values of paired selections giving entries 1,1,2,2,3 across rows/columns | M1 | |
| $\frac{7}{15}$ | A1 (AG) | |
---
## Question 5(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| $x$: 1, 2, 3; Probability: $\frac{2}{15}$, $\frac{6}{15}$, $\frac{7}{15}$ | B1 | Table structure |
| $P(1)$ or $P(2)$ correct | B1 | |
| 3rd probability correct, FT sum to 1 | B1 | |
---
## Question 5(c):
| Answer | Mark | Guidance |
|--------|------|----------|
| $E(X) = \frac{2+12+21}{15} = \frac{35}{15} = \frac{7}{3}$ | B1 | |
| $\text{Var}(X) = \frac{1^2 \times 2 + 2^2 \times 6 + 3^2 \times 7}{15} - \left(\frac{7}{3}\right)^2$ | M1 | |
| $\frac{22}{45}$ $(0.489)$ | A1 | |
---5 A fair three-sided spinner has sides numbered 1, 2, 3. A fair five-sided spinner has sides numbered $1,1,2,2,3$. Both spinners are spun once. For each spinner, the number on the side on which it lands is noted. The random variable $X$ is the larger of the two numbers if they are different, and their common value if they are the same.
\begin{enumerate}[label=(\alph*)]
\item Show that $\mathrm { P } ( X = 3 ) = \frac { 7 } { 15 }$.\\
\includegraphics[max width=\textwidth, alt={}, center]{a3b3ebd1-db9e-4552-9abe-bfdeba786d02-08_69_1569_541_328}
\item Draw up the probability distribution table for $X$.
\item Find $\mathrm { E } ( X )$ and $\operatorname { Var } ( X )$.
\end{enumerate}
\hfill \mbox{\textit{CAIE S1 2020 Q5 [8]}}