| Exam Board | OCR |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2013 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Discrete Probability Distributions |
| Type | Three or more independent values |
| Difficulty | Moderate -0.8 This is a straightforward S1 question testing standard probability distribution calculations. Part (i) requires routine application of E(X) and Var(X) formulas with simple arithmetic. Part (ii) involves basic enumeration of integer solutions and conditional probability. Part (iii) is a direct binomial probability calculation. All parts are textbook exercises with no problem-solving insight required, making this easier than average. |
| 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 | 3 | 5 | 7 |
| P\((X = x)\) | 0.4 | 0.3 | 0.2 | 0.1 |
### (i)
**Answer:**
- $1\times 0.4 + 3\times 0.3 + 5\times 0.2 + 7\times 0.1 = 3$
- $1^2\times 0.4 + 3^2\times 0.3 + 5^2\times 0.2 + 7^2\times 0.1 = "3"^2$ = 4
**Marks:** M1, A1, M1, M1, A1[5]
**Guidance:**
- $\geq 3$ terms correct + eg 4 M0
- Use of $\Sigma(x - \bar{x})^2\times p$: $2^2\times 0.4 + 0 + 2^2\times 0.2 + 4^2\times 0.1$ or 2 correct non-zero terms M1
- Dep +ve result M1
### (ii)
**Answer:** $775,\ 757,\ 577$
**Marks:** B1
**Guidance:**
- Must show all three
- Allow repeats, eg list of 6 orders
- Alt method: $X_1: 5$ or $7$, $X_2: 5$ or $7$, $X_3: 5$ or $7$ or $X_1, X_2, X_3$ can be 5 or 7 B1
### (iii)
**Answer:**
- Binomial stated, or seen or implied with any $n$ & $p$
- $^{11}C_4 \times 0.8^7 \times 0.2^4 = 0.111$ (3 sf)
**Marks:** B1, M1, A1[3]
**Guidance:**
- eg by $0.8' \times 0.2'$ (r.s.>1) not just by "C"
- Correct method
- Correct answer, no working M1M1A1
- NB 0.0388 scores B1M0A0 as it is $^{11}C_5\times 0.8^6\times 0.8^2$
---The probability distribution of a random variable $X$ is shown.
\begin{tabular}{|c|c|c|c|c|}
\hline
$x$ & 1 & 3 & 5 & 7 \\
\hline
P$(X = x)$ & 0.4 & 0.3 & 0.2 & 0.1 \\
\hline
\end{tabular}
\begin{enumerate}[label=(\roman*)]
\item Find E$(X)$ and Var$(X)$. [5]
\item Three independent values of $X$, denoted by $X_1$, $X_2$ and $X_3$, are chosen. Given that $X_1 + X_2 + X_3 = 19$, write down all the possible sets of values for $X_1$, $X_2$ and $X_3$ and hence find P$(X_1 = 7)$. [2]
\item 11 independent values of $X$ are chosen. Use an appropriate formula to find the probability that exactly 4 of these values are 5s. [3]
\end{enumerate}
\hfill \mbox{\textit{OCR S1 2013 Q3 [10]}}