| Exam Board | OCR MEI |
|---|---|
| Module | S1 (Statistics 1) |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Independent Events |
| Type | Venn diagram with independence constraint |
| Difficulty | Standard +0.3 This is a straightforward application of independence and Venn diagram probability rules. Part (i) uses the independence formula directly, part (ii) is routine diagram completion using given probabilities, and parts (iii-iv) involve standard checks of independence conditions. The question requires multiple steps but uses only basic S1 concepts with no novel problem-solving insight needed. |
| Spec | 2.03a Mutually exclusive and independent events2.03c Conditional probability: using diagrams/tables2.03d Calculate conditional probability: from first principles |
| Answer | Marks | Guidance |
|---|---|---|
| \(P(A \cap B) = P(A)P(B | A) = \frac{7}{10} \times \frac{3}{7}\) | |
| \(\rightarrow P(A \cap B) = 0.3\) | M1 product of these fractions, A1 | 2 marks |
| Answer | Marks | Guidance |
|---|---|---|
| Venn diagram with two intersecting circles labelled A and B, containing values: 0.4 (A only), 0.3 (intersection), 0.2 (B only), 0.1 (outside) | B1 FT either 0.4 or 0.2 in correct place; B1 FT all correct and labelled | 2 marks |
| Answer | Marks | Guidance |
|---|---|---|
| \(P(B | A) \neq P(B)\), \(\frac{3}{7} \neq 0.5\) | |
| Unequal so not independent | E1 correct comparison; E1 dep for 'not independent' | 2 marks |
| Answer | Marks | Guidance |
|---|---|---|
| so Isobel is less likely to score when her parents attend | E1 for comparison; E1 dep | 2 marks |
# Question 3:
## Part (i)
$P(A \cap B) = P(A)P(B|A) = \frac{7}{10} \times \frac{3}{7}$
$\rightarrow P(A \cap B) = 0.3$ | M1 product of these fractions, A1 | 2 marks
## Part (ii)
Venn diagram with two intersecting circles labelled A and B, containing values: 0.4 (A only), 0.3 (intersection), 0.2 (B only), 0.1 (outside) | B1 FT either 0.4 or 0.2 in correct place; B1 FT all correct and labelled | 2 marks
## Part (iii)
$P(B|A) \neq P(B)$, $\frac{3}{7} \neq 0.5$
Unequal so not independent | E1 correct comparison; E1 dep for 'not independent' | 2 marks
## Part (iv)
$\frac{3}{7} < 0.5$
so Isobel is less likely to score when her parents attend | E1 for comparison; E1 dep | 2 marks
---3 Isobel plays football for a local team. Sometimes her parents attend matches to watch her play.
\begin{itemize}
\item $A$ is the event that Isobel's parents watch a match.
\item $\quad B$ is the event that Isobel scores in a match.
\end{itemize}
You are given that $\frac { 3 } { 7 }$ and $\mathrm { P } ( A ) = \frac { 7 } { 10 }$.\\
(i) Calculate $\mathrm { P } ( A \cap B )$.
The probability that Isobel does not score and her parents do not attend is 0.1 .\\
(ii) Draw a Venn diagram showing the events $A$ and $B$, and mark in the probability corresponding to each of the regions of your diagram.\\
(iii) Are events $A$ and $B$ independent? Give a reason for your answer.\\
(iv) By comparing $\mathrm { P } ( B \mid A )$ with $\mathrm { P } ( B )$, explain why Isobel should ask her parents not to attend.
\hfill \mbox{\textit{OCR MEI S1 Q3 [8]}}