| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2021 |
| Session | October |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Continuous Uniform Random Variables |
| Type | Breaking/cutting problems |
| Difficulty | Standard +0.8 This S2 question combines uniform distribution theory with a novel pasta-cutting context requiring geometric insight. Part (i) involves routine manipulation of uniform distribution properties, but part (ii) requires students to recognize that the shortest piece has a non-trivial distribution (uniform on [0, 22.5]), then apply binomial probability. The conceptual leap from cutting position to shortest piece length, plus the multi-stage reasoning in (ii)(c), elevates this above standard textbook exercises. |
| Spec | 2.04b Binomial distribution: as model B(n,p)2.04c Calculate binomial probabilities5.03a Continuous random variables: pdf and cdf5.03b Solve problems: using pdf5.03c Calculate mean/variance: by integration |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(P(X > 14) = \frac{2}{5}\) oe | B1 | Allow 0.4 |
| (1) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(a = 8 - 2(14-8) = -4\) | M1 | Correct method to find \(a\), or \(\frac{a+b}{2} = 11\). May be awarded in part (a) |
| \(b = 14 + 2(14-8) = 26\) | M1 | Correct method to find \(b\), or second correct equation ft their (a), e.g. \(\frac{b-14}{b-a} = \frac{2}{5}\). May be awarded in part (a) |
| \(P(6X > a + b) = \frac{26 - \frac{26-4}{6}}{26+4}\) oe | M1 | Correct probability expression using their \(a\) and their \(b\) |
| \(= \frac{67}{90}\) oe | A1 | awrt 0.744. Correct answer |
| (4) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(S \sim U[0, 22.5]\) or \(f(s) = \begin{cases} \frac{2}{45} & 0 \leq s \leq 22.5 \\ 0 & \text{otherwise} \end{cases}\) | B1 | Correct distribution stated, allow in words. Condone \(<\) |
| (1) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(P(S < 12) = \frac{12}{"22.5"}\) | M1 | Correct method ft their value of \((b-a)\) if positive. Condone 45 in denominator |
| \(= \frac{8}{15}\) | A1 | awrt 0.533 |
| (2) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(P(T=6) = \binom{20}{6}\left(\frac{8}{15}\right)^6\left(1 - \frac{8}{15}\right)^{14}\) | M1M1 | First M1: for \(\left(\frac{"8"}{15}\right)^6\left("1-\frac{8}{15}"\right)^{14}\); Second M1: fully correct probability ft their 8/15 |
| \(= 0.02072\ldots\) | A1 | awrt 0.0207 |
| (3) | ||
| Total 11 |
# Question 2:
## Part (i)(a)
| Answer | Mark | Guidance |
|--------|------|----------|
| $P(X > 14) = \frac{2}{5}$ oe | B1 | Allow 0.4 |
| | **(1)** | |
## Part (i)(b)
| Answer | Mark | Guidance |
|--------|------|----------|
| $a = 8 - 2(14-8) = -4$ | M1 | Correct method to find $a$, or $\frac{a+b}{2} = 11$. May be awarded in part (a) |
| $b = 14 + 2(14-8) = 26$ | M1 | Correct method to find $b$, or second correct equation ft their (a), e.g. $\frac{b-14}{b-a} = \frac{2}{5}$. May be awarded in part (a) |
| $P(6X > a + b) = \frac{26 - \frac{26-4}{6}}{26+4}$ oe | M1 | Correct probability expression using their $a$ and their $b$ |
| $= \frac{67}{90}$ oe | A1 | awrt 0.744. Correct answer |
| | **(4)** | |
## Part (ii)(a)
| Answer | Mark | Guidance |
|--------|------|----------|
| $S \sim U[0, 22.5]$ or $f(s) = \begin{cases} \frac{2}{45} & 0 \leq s \leq 22.5 \\ 0 & \text{otherwise} \end{cases}$ | B1 | Correct distribution stated, allow in words. Condone $<$ |
| | **(1)** | |
## Part (ii)(b)
| Answer | Mark | Guidance |
|--------|------|----------|
| $P(S < 12) = \frac{12}{"22.5"}$ | M1 | Correct method ft their value of $(b-a)$ if positive. Condone 45 in denominator |
| $= \frac{8}{15}$ | A1 | awrt 0.533 |
| | **(2)** | |
## Part (ii)(c)
| Answer | Mark | Guidance |
|--------|------|----------|
| $P(T=6) = \binom{20}{6}\left(\frac{8}{15}\right)^6\left(1 - \frac{8}{15}\right)^{14}$ | M1M1 | First M1: for $\left(\frac{"8"}{15}\right)^6\left("1-\frac{8}{15}"\right)^{14}$; Second M1: fully correct probability ft their 8/15 |
| $= 0.02072\ldots$ | A1 | awrt 0.0207 |
| | **(3)** | |
| | **Total 11** | |
---2. (i) The continuous random variable $X$ is uniformly distributed over the interval $[ a , b ]$
Given that $\mathrm { P } ( 8 < X < 14 ) = \frac { 1 } { 5 }$ and $\mathrm { E } ( X ) = 11$
\begin{enumerate}[label=(\alph*)]
\item write down $\mathrm { P } ( X > 14 )$
\item find $\mathrm { P } ( 6 X > a + b )$\\
(ii) Susie makes a strip of pasta 45 cm long. She then cuts the strip of pasta, at a randomly chosen point, into two pieces. The random variable $S$ is the length of the shortest piece of pasta.\\
(a) Write down the distribution of $S$\\
(b) Calculate the probability that the shortest piece of pasta is less than 12 cm long.
Susie makes 20 strips of pasta, all 45 cm long, and separately cuts each strip of pasta, at a randomly chosen point, into two pieces.
\item Calculate the probability that exactly 6 of the pieces of pasta are less than 12 cm long.
\end{enumerate}
\hfill \mbox{\textit{Edexcel S2 2021 Q2 [11]}}