| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2011 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Continuous Uniform Random Variables |
| Type | Maximum or minimum of uniforms |
| Difficulty | Standard +0.3 This is a straightforward S2 question on continuous uniform distributions. Part (a) is basic probability calculation from a uniform distribution. Part (b) requires understanding that P(max > 9.5) = 1 - P(all ≤ 9.5) = 1 - (5/6)³, which is a standard technique. Part (c) is a binomial probability calculation. All parts use routine methods with no novel insight required, 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 |
| \(\frac{9.5-7}{10-7} = \frac{5}{6}\) | M1 | For an expression for the probability e.g. \(\int_7^{9.5} \frac{1}{3}\,dx\) |
| \(= \frac{5}{6}\) awrt 0.833 | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(P(\text{Longest} > 9.5) = 1 - P(\text{all} < 9.5) = 1 - \left(\frac{5}{6}\right)^3\) | M1 | For \(1-(a)^3\) or \((1-a)^3 + 3(1-a)^2a + 3(1-a)a^2\) |
| \(= \frac{91}{216}\) or 0.421 | A1 | awrt 0.421 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(P(\text{stick} < 7.6) = \frac{0.6}{3} = 0.2\) | B1 | 0.2 may be implied by at least one correct probability |
| Let \(Y\) = number of sticks (out of 6) \(<7.6\), then \(Y \sim B(6, 0.2)\) | M1 | \(1^\text{st}\) M1 for writing or using \(B(6,p)\), implied by \(np^x(1-p)^{6-x}\) with \(n \geq 1\) |
| \(P(Y>4) = 1 - P(Y \leq 4) = 1 - 0.9984\) | M1 | \(2^\text{nd}\) M1 for writing \(1 - P(Y \leq 4)\) or \(np^5(1-p)+p^6\) (\(n\) integer \(> 1\)) |
| \(= 0.0016\) or \(\frac{1}{625}\) | A1 | cao; NB 0.0016 with no working gets B0M0M0A0 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(1^\text{st}\) B1: for identifying negative skew | B1 | |
| \(2^\text{nd}\) B1: correct deduction \(E(X) < 3\) | B1 | Dependent on previous B mark |
## Question 4:
### Part (a):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\frac{9.5-7}{10-7} = \frac{5}{6}$ | M1 | For an expression for the probability e.g. $\int_7^{9.5} \frac{1}{3}\,dx$ |
| $= \frac{5}{6}$ awrt 0.833 | A1 | |
### Part (b):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $P(\text{Longest} > 9.5) = 1 - P(\text{all} < 9.5) = 1 - \left(\frac{5}{6}\right)^3$ | M1 | For $1-(a)^3$ or $(1-a)^3 + 3(1-a)^2a + 3(1-a)a^2$ |
| $= \frac{91}{216}$ or 0.421 | A1 | awrt 0.421 |
### Part (c):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $P(\text{stick} < 7.6) = \frac{0.6}{3} = 0.2$ | B1 | 0.2 may be implied by at least one correct probability |
| Let $Y$ = number of sticks (out of 6) $<7.6$, then $Y \sim B(6, 0.2)$ | M1 | $1^\text{st}$ M1 for writing or using $B(6,p)$, implied by $np^x(1-p)^{6-x}$ with $n \geq 1$ |
| $P(Y>4) = 1 - P(Y \leq 4) = 1 - 0.9984$ | M1 | $2^\text{nd}$ M1 for writing $1 - P(Y \leq 4)$ or $np^5(1-p)+p^6$ ($n$ integer $> 1$) |
| $= 0.0016$ or $\frac{1}{625}$ | A1 | cao; NB 0.0016 with no working gets B0M0M0A0 |
### Part (d):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $1^\text{st}$ B1: for identifying negative skew | B1 | |
| $2^\text{nd}$ B1: correct deduction $E(X) < 3$ | B1 | Dependent on previous B mark |
---\begin{enumerate}
\item In a game, players select sticks at random from a box containing a large number of sticks of different lengths. The length, in cm , of a randomly chosen stick has a continuous uniform distribution over the interval [7,10].
\end{enumerate}
A stick is selected at random from the box.\\
(a) Find the probability that the stick is shorter than 9.5 cm .
To win a bag of sweets, a player must select 3 sticks and wins if the length of the longest stick is more than 9.5 cm .\\
(b) Find the probability of winning a bag of sweets.
To win a soft toy, a player must select 6 sticks and wins the toy if more than four of the sticks are shorter than 7.6 cm .\\
(c) Find the probability of winning a soft toy.\\
\hfill \mbox{\textit{Edexcel S2 2011 Q4 [8]}}