| Exam Board | OCR |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2010 |
| Session | January |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Discrete Probability Distributions |
| Type | Calculate Var(X) from table |
| Difficulty | Moderate -0.3 This is a straightforward S1 question testing standard probability distribution calculations (E(X) and Var(X) using formulas) and binomial probability applications. While it requires multiple steps and careful arithmetic with fractions, all techniques are routine textbook exercises with no problem-solving insight needed—slightly easier than average due to its mechanical nature. |
| 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 variables5.02c Linear coding: effects on mean and variance |
| \(x\) | 0 | 1 | 2 | 3 |
| P\((X = x)\) | \(\frac{1}{2}\) | \(\frac{1}{4}\) | \(\frac{1}{8}\) | \(\frac{1}{8}\) |
| Answer | Marks | Guidance |
|---|---|---|
| \((0x\frac{1}{4}) + 1x\frac{1}{4} + 2x\frac{1}{8} + 3x\frac{1}{8} = \frac{7}{8}\) or \(0.875\) oe | M1 A1 | ≥ 2 non-zero terms seen; If =3 or 4 M0M0M1(poss) ≥ 2 non-zero terms seen |
| \((0x\frac{1}{4}) + 1x\frac{1}{4} + 2^2x\frac{1}{8} + 3^2x\frac{1}{8}\) (= \(1\frac{7}{8}\)) | M1 | dep +ve result M1 all4 \((x-0.875)^2\) terms seen. M1 mult p.\(\sum\) A1 1.11 |
| \(-(\frac{7}{8})^2 = \frac{71}{64}\) or \(1.11\) (3 sfs) oe | A1 5 |
| Answer | Marks | Guidance |
|---|---|---|
| Bin stated or implied \(0.922\) (3 sfs) | M1 A1 2 | Eg table or \(\frac{1}{4} \times \frac{3}{4}\) \((n+m=10,n,m\neq 1)\) or 10C4 or 5(or 4 or 6) terms correct |
| Answer | Marks | Guidance |
|---|---|---|
| \(^{10}C_4x\frac{3}{8}^6 \times\frac{1}{8}^4 = 0.0230\) (3 sfs) | M1 A1 3 | condone 0.023 |
### (i)
$(0x\frac{1}{4}) + 1x\frac{1}{4} + 2x\frac{1}{8} + 3x\frac{1}{8} = \frac{7}{8}$ or $0.875$ oe | M1 A1 | ≥ 2 non-zero terms seen; If =3 or 4 M0M0M1(poss) ≥ 2 non-zero terms seen
$(0x\frac{1}{4}) + 1x\frac{1}{4} + 2^2x\frac{1}{8} + 3^2x\frac{1}{8}$ (= $1\frac{7}{8}$) | M1 | dep +ve result M1 all4 $(x-0.875)^2$ terms seen. M1 mult p.$\sum$ A1 1.11
$-(\frac{7}{8})^2 = \frac{71}{64}$ or $1.11$ (3 sfs) oe | A1 5 |
### (ii)
Bin stated or implied $0.922$ (3 sfs) | M1 A1 2 | Eg table or $\frac{1}{4} \times \frac{3}{4}$ $(n+m=10,n,m\neq 1)$ or 10C4 or 5(or 4 or 6) terms correct
### (iii)
$n = 10$ & $p = \frac{3}{8}$ stated or implied
$^{10}C_4x\frac{3}{8}^6 \times\frac{1}{8}^4 = 0.0230$ (3 sfs) | M1 A1 3 | condone 0.023
**Total [10]**
---A certain four-sided die is biased. The score, $X$, on each throw is a random variable with probability distribution as shown in the table. Throws of the die are independent.
\begin{tabular}{|c|c|c|c|c|}
\hline
$x$ & 0 & 1 & 2 & 3 \\
\hline
P$(X = x)$ & $\frac{1}{2}$ & $\frac{1}{4}$ & $\frac{1}{8}$ & $\frac{1}{8}$ \\
\hline
\end{tabular}
\begin{enumerate}[label=(\roman*)]
\item Calculate E$(X)$ and Var$(X)$. [5]
\end{enumerate}
The die is thrown 10 times.
\begin{enumerate}[label=(\roman*)]
\setcounter{enumi}{1}
\item Find the probability that there are not more than 4 throws on which the score is 1. [2]
\item Find the probability that there are exactly 4 throws on which the score is 2. [3]
\end{enumerate}
\hfill \mbox{\textit{OCR S1 2010 Q4 [10]}}