| Exam Board | OCR MEI |
|---|---|
| Module | S1 (Statistics 1) |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Conditional Probability |
| Type | Independence test with conditional probability |
| Difficulty | Moderate -0.8 This is a straightforward application of basic conditional probability and independence definitions. Part (i) requires comparing P(L|R) with P(L), part (ii) uses the formula P(L∩R) = P(L|R)×P(R), and part (iii) is routine Venn diagram completion. All steps are direct recall of standard formulas with minimal calculation, making this easier than average. |
| Spec | 2.03a Mutually exclusive and independent events2.03b Probability diagrams: tree, Venn, sample space2.03c Conditional probability: using diagrams/tables2.03d Calculate conditional probability: from first principles |
| Answer | Marks | Guidance |
|---|---|---|
| Because \(P(L \mid R) \neq P(L)\) | E1 [1] | Allow \(0.45 \neq 0.15\); Either \(P(L \cap R) = 0.099 \neq P(L) \times P(R)\); or \(0.099 \neq 0.15 \times 0.22 = 0.033\) |
| Answer | Marks | Guidance |
|---|---|---|
| \(P(L \cap R) = P(L \mid R) \times P(R) = 0.45 \times 0.22 = 0.099\) | M1, A1 [2] | For product; CAO; Allow \(\frac{99}{1000}\) |
| Answer | Marks | Guidance |
|---|---|---|
| Venn diagram with two intersecting circles labelled \(L\) and \(R\), showing values: 0.051, 0.099, 0.121, 0.729 | G1, G1, G1 [3] | G1 for two labelled intersecting circles; G1 for at least 2 correct probabilities, FT their \(P(L \cap R)\) from (ii) provided \(\leq 0.15\); G1 for remaining probabilities, FT their \(P(L \cap R)\) providing probabilities between 0 and 1 |
## Question 4:
**(i)**
Because $P(L \mid R) \neq P(L)$ | E1 [1] | Allow $0.45 \neq 0.15$; Either $P(L \cap R) = 0.099 \neq P(L) \times P(R)$; or $0.099 \neq 0.15 \times 0.22 = 0.033$
**(ii)**
$P(L \cap R) = P(L \mid R) \times P(R) = 0.45 \times 0.22 = 0.099$ | M1, A1 [2] | For product; CAO; Allow $\frac{99}{1000}$
**(iii)**
Venn diagram with two intersecting circles labelled $L$ and $R$, showing values: 0.051, 0.099, 0.121, 0.729 | G1, G1, G1 [3] | G1 for two labelled intersecting circles; G1 for at least 2 correct probabilities, FT their $P(L \cap R)$ from (ii) provided $\leq 0.15$; G1 for remaining probabilities, FT their $P(L \cap R)$ providing probabilities between 0 and 1
---4 Each weekday, Marta travels to school by bus. Sometimes she arrives late.
\begin{itemize}
\item $L$ is the event that Marta arrives late.
\item $R$ is the event that it is raining.
\end{itemize}
You are given that $\mathrm { P } ( L ) = 0.15 , \mathrm { P } ( R ) = 0.22$ and $\mathrm { P } ( L \mid R ) = 0.45$.\\
(i) Use this information to show that the events $L$ and $R$ are not independent.\\
(ii) Find $\mathrm { P } ( L \cap R )$.\\
(iii) Draw a Venn diagram showing the events $L$ and $R$, and fill in the probability corresponding to each of the four regions of your diagram.
\hfill \mbox{\textit{OCR MEI S1 Q4 [6]}}