| Exam Board | OCR |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2016 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Discrete Probability Distributions |
| Type | Three or more independent values |
| Difficulty | Moderate -0.3 This is a straightforward S1 question requiring standard calculations: (i) uses basic expectation and variance formulas from a given distribution table, and (ii) applies binomial probability with n=3 trials. Both parts are routine textbook exercises with no conceptual challenges, making it slightly easier than average but not trivial due to the multi-step variance calculation and binomial reasoning required. |
| Spec | 2.04b Binomial distribution: as model B(n,p)2.04c Calculate binomial probabilities5.02a Discrete probability distributions: general5.02b Expectation and variance: discrete random variables |
| \(x\) | 1 | 2 | 3 | 4 |
| \(\mathrm { P } ( X = x )\) | 0.1 | 0.3 | 0.4 | 0.2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\Sigma xp = 2.7\) oe | M1, A1 | \(\geq 3\) correct terms; \(\div 4\) or \(\div 10\) etc M0A0 |
| \(\Sigma x^2 p = 8.1\) | M1 | \(\geq 3\) correct terms; \((x - 2.7')^2 \geq 3\) correct terms M1 |
| \(- \text{"2.7"}^2\) | M1 | dep +ve result; \(\Sigma(x-\text{"2.7"})^2p \geq 3\) correct terms M1 |
| \(= 0.81\) oe | A1 [5] | \(\div 4\) or \(\div 10\) etc M0M0 A0 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(0.3^2 \times 0.7 \times 3 + 0.3^3\) or \(0.189 + 0.027\) oe | M2 | or \(0.3^2 \times 0.7 + 0.3^3\) M1; or \(0.3^2 \times 0.7 \times 3\) or \(0.189\) oe M1; \(1-(^3C_1 \times 0.3 \times 0.7^2 + 0.7^3)\) M2; or \(1-(0.3 \times 0.7^2 + 0.7^3)\) M1 |
| \(= \frac{27}{125}\) or \(0.216\) | A1 [3] | \(0.3^2 \times 0.1 \times 3 + 0.3^2 \times 0.4 \times 3 + 0.3^2 \times 0.2 \times 3 + 0.3^3\) M2; SC: M1 for \(0.3^2 \times 0.6 \times 3 + 0.3^3\); \(\times^3C_1\) or \(\times^3C_2\) instead of \(\times 3\) is OK throughout |
# Question 1:
## Part (i)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\Sigma xp = 2.7$ oe | M1, A1 | $\geq 3$ correct terms; $\div 4$ or $\div 10$ etc M0A0 |
| $\Sigma x^2 p = 8.1$ | M1 | $\geq 3$ correct terms; $(x - 2.7')^2 \geq 3$ correct terms M1 |
| $- \text{"2.7"}^2$ | M1 | dep +ve result; $\Sigma(x-\text{"2.7"})^2p \geq 3$ correct terms M1 |
| $= 0.81$ oe | A1 **[5]** | $\div 4$ or $\div 10$ etc M0M0 A0 |
## Part (ii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $0.3^2 \times 0.7 \times 3 + 0.3^3$ or $0.189 + 0.027$ oe | M2 | or $0.3^2 \times 0.7 + 0.3^3$ M1; or $0.3^2 \times 0.7 \times 3$ or $0.189$ oe M1; $1-(^3C_1 \times 0.3 \times 0.7^2 + 0.7^3)$ M2; or $1-(0.3 \times 0.7^2 + 0.7^3)$ M1 |
| $= \frac{27}{125}$ or $0.216$ | A1 **[3]** | $0.3^2 \times 0.1 \times 3 + 0.3^2 \times 0.4 \times 3 + 0.3^2 \times 0.2 \times 3 + 0.3^3$ M2; SC: M1 for $0.3^2 \times 0.6 \times 3 + 0.3^3$; $\times^3C_1$ or $\times^3C_2$ instead of $\times 3$ is OK throughout |
---1 The table shows the probability distribution of a random variable $X$.
\begin{center}
\begin{tabular}{ | c | c | c | c | c | }
\hline
$x$ & 1 & 2 & 3 & 4 \\
\hline
$\mathrm { P } ( X = x )$ & 0.1 & 0.3 & 0.4 & 0.2 \\
\hline
\end{tabular}
\end{center}
(i) Find $\mathrm { E } ( X )$ and $\operatorname { Var } ( X )$.\\
(ii) Three values of $X$ are chosen at random. Find the probability that $X$ takes the value 2 at least twice.
\hfill \mbox{\textit{OCR S1 2016 Q1 [8]}}