| Exam Board | Edexcel |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2001 |
| Session | June |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Principle of Inclusion/Exclusion |
| Type | Standard Survey to Venn Diagram |
| Difficulty | Easy -1.2 This is a straightforward application of inclusion-exclusion with all values directly given. Students simply fill in a Venn diagram systematically (working from the center outward) and read off probabilities. It requires careful arithmetic but no problem-solving insight or novel reasoning—purely mechanical execution of a standard S1 technique. |
| Spec | 2.03b Probability diagrams: tree, Venn, sample space |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Venn diagram with \(2\) in A only region | B1 | \(2\) |
| \(4, 3, 5\) in correct regions | M1A1 | |
| \(21, 16, 10\) in correct regions | M1A1 | |
| \(39\) outside all circles | B1 (6) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(P(\text{at least one}) = \frac{21+3+\cdots+10}{100}\) or \(1 - \frac{39}{100}\) | M1 | |
| \(= \frac{61}{100} = \underline{0.61}\) | A1\(\checkmark\) (2) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(P(\text{only } A) = \frac{21}{100} = \underline{0.21}\) | B1\(\checkmark\) (1) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(P(\text{only one}) = \frac{21+16+10}{100}\) | M1 | |
| \(= \frac{47}{100} = \underline{0.47}\) | A1\(\checkmark\) (2) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(P(A \mid \text{only reads one}) = \frac{0.21}{0.47}\) | M1 | Use of \(\frac{P(A \cap B)}{P(E)}\); i.e. their (c)/their (d) |
| \(= \frac{21}{47} = \underline{0.4468\ldots}\) | A1\(\checkmark\) (2) | AWRT \(0.45\) |
## Question 5:
### Part (a)
| Answer/Working | Mark | Guidance |
|---|---|---|
| Venn diagram with $2$ in A only region | B1 | $2$ |
| $4, 3, 5$ in correct regions | M1A1 | |
| $21, 16, 10$ in correct regions | M1A1 | |
| $39$ outside all circles | B1 (6) | |
### Part (b)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $P(\text{at least one}) = \frac{21+3+\cdots+10}{100}$ or $1 - \frac{39}{100}$ | M1 | |
| $= \frac{61}{100} = \underline{0.61}$ | A1$\checkmark$ (2) | |
### Part (c)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $P(\text{only } A) = \frac{21}{100} = \underline{0.21}$ | B1$\checkmark$ (1) | |
### Part (d)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $P(\text{only one}) = \frac{21+16+10}{100}$ | M1 | |
| $= \frac{47}{100} = \underline{0.47}$ | A1$\checkmark$ (2) | |
### Part (e)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $P(A \mid \text{only reads one}) = \frac{0.21}{0.47}$ | M1 | Use of $\frac{P(A \cap B)}{P(E)}$; i.e. their (c)/their (d) |
| $= \frac{21}{47} = \underline{0.4468\ldots}$ | A1$\checkmark$ (2) | AWRT $0.45$ |
---5. A market researcher asked 100 adults which of the three newspapers $A , B , C$ they read. The results showed that $30 \operatorname { read } A , 26$ read $B , 21$ read $C , 5$ read both $A$ and $B , 7$ read both $B$ and $C , 6$ read both $C$ and $A$ and 2 read all three.
\begin{enumerate}[label=(\alph*)]
\item Draw a Venn diagram to represent these data.
One of the adults is then selected at random.\\
Find the probability that she reads
\item at least one of the newspapers,
\item only $A$,
\item only one of the newspapers,
\item $A$ given that she reads only one newspaper.
\end{enumerate}
\hfill \mbox{\textit{Edexcel S1 2001 Q5 [13]}}