| Exam Board | Edexcel |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2005 |
| Session | January |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Principle of Inclusion/Exclusion |
| Type | Standard Survey to Venn Diagram |
| Difficulty | Moderate -0.8 This is a straightforward application of inclusion-exclusion with all values given directly. Part (a) requires filling in a standard 3-set Venn diagram by working backwards from intersections. Parts (b)-(e) involve basic probability calculations from the completed diagram with no conceptual challenges—purely mechanical arithmetic once the diagram is drawn. |
| Spec | 2.03a Mutually exclusive and independent events2.03b Probability diagrams: tree, Venn, sample space2.03c Conditional probability: using diagrams/tables |
| Answer | Marks | Guidance |
|---|---|---|
| Venn diagram with values: 16, 5, 19, 6, 4, 7, 25, 918 | B1 | 6 |
| M1 | subtract | |
| A1 | 4, 5, 7 | |
| A1 | subtract | |
| A1 | 16, 19, 25 | |
| B1 | 918 |
| Answer | Marks |
|---|---|
| \(P(\text{No defects}) = \frac{918}{1000} = 0.918\) | B1 |
| Answer | Marks |
|---|---|
| \(P(\text{No more than 1}) = \frac{918+16+19+25}{1000}\) OR \(1 - \frac{5+6+4+7}{1000}\) | M1 |
| \(= 0.978\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| \(P(B \mid \text{Only 1 defect}) = \frac{P(\text{B and 1 defect})}{P(\text{1 defect})} = \frac{\frac{19}{1000}}{\frac{16+19+25}{1000}}\) | M1 | conditional prob |
| \(= \frac{19}{60}\) | A1 | \(\frac{19}{60}\) or \(0.31\dot{6}\) or \(0.317\) |
| Answer | Marks | Guidance |
|---|---|---|
| \(P(\text{Both had type B}) = \frac{37}{1000} \times \frac{36}{999}\) | M1 | theirs from B \(\times\) |
| \(= \frac{37}{27750}\) or \(0.001\dot{3}\) or \(0.00133\) | A1 | cao |
## Question 5:
**Part (a):**
Venn diagram with values: 16, 5, 19, 6, 4, 7, 25, 918 | B1 | 6
| M1 | subtract
| A1 | 4, 5, 7
| A1 | subtract
| A1 | 16, 19, 25
| B1 | 918
**Part (b):**
$P(\text{No defects}) = \frac{918}{1000} = 0.918$ | B1 |
**Part (c):**
$P(\text{No more than 1}) = \frac{918+16+19+25}{1000}$ **OR** $1 - \frac{5+6+4+7}{1000}$ | M1 |
$= 0.978$ | A1 |
**Part (d):**
$P(B \mid \text{Only 1 defect}) = \frac{P(\text{B and 1 defect})}{P(\text{1 defect})} = \frac{\frac{19}{1000}}{\frac{16+19+25}{1000}}$ | M1 | conditional prob
$= \frac{19}{60}$ | A1 | $\frac{19}{60}$ or $0.31\dot{6}$ or $0.317$
**Part (e):**
$P(\text{Both had type B}) = \frac{37}{1000} \times \frac{36}{999}$ | M1 | theirs from B $\times$
$= \frac{37}{27750}$ or $0.001\dot{3}$ or $0.00133$ | A1 | cao
---5. Articles made on a lathe are subject to three kinds of defect, $A , B$ or $C$. A sample of 1000 articles was inspected and the following results were obtained.
\begin{displayquote}
31 had a type $A$ defect\\
37 had a type $B$ defect\\
42 had a type $C$ defect\\
11 had both type $A$ and type $B$ defects\\
13 had both type $B$ and type $C$ defects\\
10 had both type $A$ and type $C$ defects\\
6 had all three types of defect.
\begin{enumerate}[label=(\alph*)]
\item Draw a Venn diagram to represent these data.
\end{displayquote}
Find the probability that a randomly selected article from this sample had
\item no defects,
\item no more than one of these defects.
An article selected at random from this sample had only one defect.
\item Find the probability that it was a type $B$ defect.
Two different articles were selected at random from this sample.
\item Find the probability that both had type $B$ defects.
\end{enumerate}
\hfill \mbox{\textit{Edexcel S1 2005 Q5 [13]}}