| Exam Board | OCR MEI |
|---|---|
| Module | S1 (Statistics 1) |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Conditional Probability |
| Type | Independence test requiring preliminary calculations |
| Difficulty | Moderate -0.3 This is a straightforward conditional probability question requiring standard formulas: P(L∩W) = P(L|W)×P(W), then using P(L∪W) to complete a Venn diagram, and finally testing independence via P(L∩W) = P(L)×P(W). All steps are routine applications of AS-level probability formulas with no conceptual challenges or novel problem-solving required, making it slightly easier than average. |
| Spec | 2.03b Probability diagrams: tree, Venn, sample space2.03c Conditional probability: using diagrams/tables2.03d Calculate conditional probability: from first principles |
| Answer | Marks | Guidance |
|---|---|---|
| \(P(L \cap W) = P(L \mid W) \times P(W) = 0.4 \times 0.07 = 0.028\) | M1, A1 [2] | For \(P(L \mid W) \times P(W)\); cao |
| Answer | Marks | Guidance |
|---|---|---|
| Venn diagram with two intersecting circles labelled \(L\) and \(W\), showing values: 0.01, 0.028, 0.042, 0.92 | B1, B1, B1 [3] | B1 for two labelled intersecting circles; B1 for at least 2 correct probabilities, FT their 0.028 provided \(< 0.038\); B1 for remaining probabilities |
| Answer | Marks | Guidance |
|---|---|---|
| \(P(L \cap W) = 0.028\), \(P(L) \times P(W) = 0.038 \times 0.07 = 0.00266\); Not equal so not independent | M1, A1, E1* [3] | M1 for correct use of \(P(L) \times P(W)\); A1 for 0.00266; Allow 'they are dependent'; Do not award E1 if \(P(L \cap W)\) wrong; dep on M1 |
## Question 5:
**(i)**
$P(L \cap W) = P(L \mid W) \times P(W) = 0.4 \times 0.07 = 0.028$ | M1, A1 [2] | For $P(L \mid W) \times P(W)$; cao
**(ii)**
Venn diagram with two intersecting circles labelled $L$ and $W$, showing values: 0.01, 0.028, 0.042, 0.92 | B1, B1, B1 [3] | B1 for two labelled intersecting circles; B1 for at least 2 correct probabilities, FT their 0.028 provided $< 0.038$; B1 for remaining probabilities
**(iii)**
$P(L \cap W) = 0.028$, $P(L) \times P(W) = 0.038 \times 0.07 = 0.00266$; Not equal so not independent | M1, A1, E1* [3] | M1 for correct use of $P(L) \times P(W)$; A1 for 0.00266; Allow 'they are dependent'; Do not award E1 if $P(L \cap W)$ wrong; dep on M1
---5 Each weekday Alan drives to work. On his journey, he goes over a level crossing. Sometimes he has to wait at the level crossing for a train to pass.
\begin{itemize}
\item $W$ is the event that Alan has to wait at the level crossing.
\item $L$ is the event that Alan is late for work.
\end{itemize}
You are given that $\mathrm { P } ( L \mid W ) = 0.4 , \mathrm { P } ( W ) = 0.07$ and $\mathrm { P } ( L \cup W ) = 0.08$.\\
(i) Calculate $\mathrm { P } ( L \cap W )$.\\
(ii) Draw a Venn diagram, showing the events $L$ and $W$. Fill in the probability corresponding to each of the four regions of your diagram.\\
(iii) Determine whether the events $L$ and $W$ are independent, explaining your method clearly.
\hfill \mbox{\textit{OCR MEI S1 Q5 [8]}}