| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2021 |
| Session | November |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Discrete Probability Distributions |
| Type | Two unknowns from sum and expectation |
| Difficulty | Moderate -0.3 This is a standard discrete probability distribution question requiring routine application of formulas. Part (a) involves solving two simultaneous equations (sum of probabilities = 1 and E(X) = Σxp(x)), which is straightforward algebra. Parts (b)-(d) apply standard variance and binomial/geometric distribution formulas with no conceptual challenges. Slightly easier than average due to the mechanical nature of all parts. |
| Spec | 5.02b Expectation and variance: discrete random variables5.02c Linear coding: effects on mean and variance5.02f Geometric distribution: conditions |
| \(x\) | 0 | 1 | 2 | 3 |
| \(\mathrm { P } ( X = x )\) | 0.6 | \(p\) | \(q\) | 0.05 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(p + q + 0.65 = 1\) | B1 | Sum of probabilities \(= 1\). |
| \(p + 2q + 0.15 = 0.55\) | B1 | Use given information. |
| Solve 2 linear equations | M1 | Either a single expression with one variable eliminated formed or two expressions with both variables on same side seen with at least one variable value stated. |
| \(p = 0.3,\ \dfrac{3}{10};\ \ q = 0.05,\ \dfrac{1}{20}\) | A1 | CAO, both WWW. If M0 with correct answers SC B1. |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(\text{Var}(X) = \textit{their}\ 0.3 + 4\times \textit{their}\ 0.05 + 9\times0.05 - 0.55^2\) | M1 | Appropriate variance formula including \((E(X))^2\), accept unsimplified. |
| \(0.6475\ \left[\dfrac{259}{400}\right]\) | A1 | CAO (must be exact). |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(1 - P(0,1,2) = 1 - \!\left(^{12}C_0\,0.3^0\,0.7^{12} + ^{12}C_1\,0.3^1\,0.7^{11} + ^{12}C_2\,0.3^2\,0.7^{10}\right)\) | M1 | One correct term: \(^{12}C_x\,p^x\,(1-p)^{12-x}\) for \(0 < x < 12\), \(0 < p < 1\). |
| \(1 - (0.01384 + 0.07118 + 0.16779)\) | A1FT | Correct unsimplified expression, or better in final answer. Unsimplified expression must be seen to FT *their p* from 6(a) or correct. |
| \(0.747\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \((0.95)^8 \times 0.05 = 0.0332\) or \(0.95^8 - 0.95^9 = 0.0332\) | B1 | Evaluated. |
## Question 6(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| $p + q + 0.65 = 1$ | B1 | Sum of probabilities $= 1$. |
| $p + 2q + 0.15 = 0.55$ | B1 | Use given information. |
| Solve 2 linear equations | M1 | Either a single expression with one variable eliminated formed or two expressions with both variables on same side seen with at least one variable value stated. |
| $p = 0.3,\ \dfrac{3}{10};\ \ q = 0.05,\ \dfrac{1}{20}$ | A1 | CAO, both WWW. If M0 with correct answers **SC B1**. |
---
## Question 6(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| $\text{Var}(X) = \textit{their}\ 0.3 + 4\times \textit{their}\ 0.05 + 9\times0.05 - 0.55^2$ | M1 | Appropriate variance formula including $(E(X))^2$, accept unsimplified. |
| $0.6475\ \left[\dfrac{259}{400}\right]$ | A1 | CAO (must be exact). |
---
## Question 6(c):
| Answer | Mark | Guidance |
|--------|------|----------|
| $1 - P(0,1,2) = 1 - \!\left(^{12}C_0\,0.3^0\,0.7^{12} + ^{12}C_1\,0.3^1\,0.7^{11} + ^{12}C_2\,0.3^2\,0.7^{10}\right)$ | M1 | One correct term: $^{12}C_x\,p^x\,(1-p)^{12-x}$ for $0 < x < 12$, $0 < p < 1$. |
| $1 - (0.01384 + 0.07118 + 0.16779)$ | A1FT | Correct unsimplified expression, or better in final answer. Unsimplified expression must be seen to **FT** *their p* from **6(a)** or correct. |
| $0.747$ | A1 | |
---
## Question 6(d):
| Answer | Mark | Guidance |
|--------|------|----------|
| $(0.95)^8 \times 0.05 = 0.0332$ or $0.95^8 - 0.95^9 = 0.0332$ | B1 | Evaluated. |6 In a game, Jim throws three darts at a board. This is called a 'turn'. The centre of the board is called the bull's-eye.
The random variable $X$ is the number of darts in a turn that hit the bull's-eye. The probability distribution of $X$ is given in the following table.
\begin{center}
\begin{tabular}{ | c | c | c | c | c | }
\hline
$x$ & 0 & 1 & 2 & 3 \\
\hline
$\mathrm { P } ( X = x )$ & 0.6 & $p$ & $q$ & 0.05 \\
\hline
\end{tabular}
\end{center}
It is given that $\mathrm { E } ( X ) = 0.55$.
\begin{enumerate}[label=(\alph*)]
\item Find the values of $p$ and $q$.
\item Find $\operatorname { Var } ( X )$.\\
Jim is practising for a competition and he repeatedly throws three darts at the board.
\item Find the probability that $X = 1$ in at least 3 of 12 randomly chosen turns.
\item Find the probability that Jim first succeeds in hitting the bull's-eye with all three darts on his 9th turn.
\end{enumerate}
\hfill \mbox{\textit{CAIE S1 2021 Q6 [10]}}