| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2012 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Continuous Uniform Random Variables |
| Type | Multiple observations or trials |
| Difficulty | Moderate -0.3 This is a straightforward multi-part question combining basic continuous uniform distribution (simple probability calculation) with binomial distribution (standard 'at least k successes' calculation). All steps are routine applications of standard formulas with no conceptual challenges, making it slightly easier than average. |
| Spec | 5.02b Expectation and variance: discrete random variables5.02c Linear coding: effects on mean and variance5.02d Binomial: mean np and variance np(1-p)5.03a Continuous random variables: pdf and cdf |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(P(L>24) = \frac{1}{15} \times 6\) | M1 | \(\frac{1}{15}\times(6\) or \(5.5\) or \(6.5\) or \((30-24))\) or \(1-\frac{1}{15}((24-15)\) or \((23.5-15)\) or \((24.5-15))\) |
| \(= \frac{2}{5}\) or \(0.4\) oe | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Let \(X\) = number of sweets with \(L > 24\); \(X \sim B(20, 0.4)\) | M1 | Using \(B(20,\) "their (a)") |
| \(P(X \geq 8) = 1 - P(X \leq 7)\) | M1dep | Dependent on 1st M1; writing or use of \(1 - P(X \leq 7)\) |
| \(= 1 - 0.4159\) | NB Use of normal/Poisson/uniform gets M0 M0 A0 | |
| \(= 0.5841\) | A1 | awrt \(0.584\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(P(\text{both } X \geq 8) = (0.5841)^2\) | M1 | \((\text{their (b)})^2\) or \((0.58)^2\) or \((0.5841)^2\) or \((0.584)^2\) |
| \(= 0.341\ldots\) | A1 ft | Either awrt \(0.34\) or follow through on part (b); must be 2sf or better |
# Question 1:
## Part (a)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $P(L>24) = \frac{1}{15} \times 6$ | M1 | $\frac{1}{15}\times(6$ or $5.5$ or $6.5$ or $(30-24))$ or $1-\frac{1}{15}((24-15)$ or $(23.5-15)$ or $(24.5-15))$ |
| $= \frac{2}{5}$ or $0.4$ oe | A1 | |
## Part (b)
| Answer/Working | Marks | Guidance |
|---|---|---|
| Let $X$ = number of sweets with $L > 24$; $X \sim B(20, 0.4)$ | M1 | Using $B(20,$ "their (a)") |
| $P(X \geq 8) = 1 - P(X \leq 7)$ | M1dep | Dependent on 1st M1; writing or use of $1 - P(X \leq 7)$ |
| $= 1 - 0.4159$ | | NB Use of normal/Poisson/uniform gets M0 M0 A0 |
| $= 0.5841$ | A1 | awrt $0.584$ |
## Part (c)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $P(\text{both } X \geq 8) = (0.5841)^2$ | M1 | $(\text{their (b)})^2$ or $(0.58)^2$ or $(0.5841)^2$ or $(0.584)^2$ |
| $= 0.341\ldots$ | A1 ft | Either awrt $0.34$ or follow through on part (b); must be 2sf or better |
---\begin{enumerate}
\item A manufacturer produces sweets of length $L \mathrm {~mm}$ where $L$ has a continuous uniform distribution with range [15, 30].\\
(a) Find the probability that a randomly selected sweet has a length greater than 24 mm .
\end{enumerate}
These sweets are randomly packed in bags of 20 sweets.\\
(b) Find the probability that a randomly selected bag will contain at least 8 sweets with length greater than 24 mm .\\
(c) Find the probability that 2 randomly selected bags will both contain at least 8 sweets with length greater than 24 mm .\\
\hfill \mbox{\textit{Edexcel S2 2012 Q1 [7]}}