| Exam Board | OCR MEI |
|---|---|
| Module | S1 (Statistics 1) |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Principle of Inclusion/Exclusion |
| Type | Three-Set Venn Diagram Probability Calculation |
| Difficulty | Moderate -0.8 This is a straightforward Venn diagram reading exercise requiring basic probability calculations. Part (i) involves simple counting from the diagram and dividing by 100, while part (ii) uses the complement rule with a standard binomial probability calculation. No problem-solving insight is needed—just careful arithmetic and routine probability formulas. |
| Spec | 2.03b Probability diagrams: tree, Venn, sample space2.04b Binomial distribution: as model B(n,p) |
| Answer | Marks | Guidance |
|---|---|---|
| \(P(\text{at most one}) = \frac{83}{100} = 0.83\) | B1 aef | 1 mark |
| Answer | Marks | Guidance |
|---|---|---|
| \(P(\text{exactly two}) = \frac{10+2+1}{100} = \frac{13}{100} = 0.13\) | M1 for \((10+2+1)/100\); A1 aef | 2 marks |
| Answer | Marks | Guidance |
|---|---|---|
| \(P(\text{all at least one}) = \frac{53}{100} \times \frac{52}{99} \times \frac{51}{98} = \frac{140556}{970200} = 0.145\) | M1 for \(\frac{53}{100} \times\); M1 dep for product of next 2 correct fractions; A1 CAO | 3 marks |
## Question 6:
### Part (i)(A)
$P(\text{at most one}) = \frac{83}{100} = 0.83$ | B1 aef | **1 mark**
### Part (i)(B)
$P(\text{exactly two}) = \frac{10+2+1}{100} = \frac{13}{100} = 0.13$ | M1 for $(10+2+1)/100$; A1 aef | **2 marks**
### Part (ii)
$P(\text{all at least one}) = \frac{53}{100} \times \frac{52}{99} \times \frac{51}{98} = \frac{140556}{970200} = 0.145$ | M1 for $\frac{53}{100} \times$; M1 dep for product of next 2 correct fractions; A1 CAO | **3 marks**6 Whitefly, blight and mosaic virus are three problems which can affect tomato plants. 100 tomato plants are examined for these problems. The numbers of plants with each type of problem are shown in the Venn diagram. 47 of the plants have none of the problems.\\
\includegraphics[max width=\textwidth, alt={}, center]{e54eba7c-d862-435a-acdd-27df6ede5fab-3_654_804_1262_699}
\begin{enumerate}[label=(\roman*)]
\item One of the 100 plants is selected at random. Find the probability that this plant has\\
(A) at most one of the problems,\\
(B) exactly two of the problems.
\item Three of the 100 plants are selected at random. Find the probability that all of them have at least one of the problems.
\end{enumerate}
\hfill \mbox{\textit{OCR MEI S1 Q6 [6]}}