| Exam Board | OCR MEI |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2010 |
| Session | January |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Binomial Distribution |
| Type | Single batch expected count |
| Difficulty | Moderate -0.8 This is a straightforward application of the binomial distribution requiring only direct formula substitution. Part (i) asks for P(X=20) where X~B(30,0.6), and part (ii) simply multiplies this probability by 100. No problem-solving or conceptual insight is needed beyond recognizing the binomial model, making it easier than average. |
| Spec | 2.04b Binomial distribution: as model B(n,p)2.04c Calculate binomial probabilities |
| Answer | Marks | Guidance |
|---|---|---|
| \(P(20 \text{ correct}) = \binom{30}{20} \times 0.6^{20} \times 0.4^{10} = 0.1152\) | M1 for \(0.6^{20} \times 0.4^{10}\), M1 for \(\binom{30}{20} \times p^{20}q^{10}\), A1 CAO | [3] |
| Answer | Marks | Guidance |
|---|---|---|
| Expected number \(= 100 \times 0.1152 = 11.52\) | M1, A1 FT (Must not round to whole number) | [2] |
| TOTAL | [5] |
## (i)
$P(20 \text{ correct}) = \binom{30}{20} \times 0.6^{20} \times 0.4^{10} = 0.1152$ | M1 for $0.6^{20} \times 0.4^{10}$, M1 for $\binom{30}{20} \times p^{20}q^{10}$, A1 CAO | [3]
## (ii)
Expected number $= 100 \times 0.1152 = 11.52$ | M1, A1 FT (Must not round to whole number) | [2]
**TOTAL** | | [5]
---In a multiple-choice test there are 30 questions. For each question, there is a 60% chance that a randomly selected student answers correctly, independently of all other questions.
\begin{enumerate}[label=(\roman*)]
\item Find the probability that a randomly selected student gets a total of exactly 20 questions correct. [3]
\item If 100 randomly selected students take the test, find the expected number of students who get exactly 20 questions correct. [2]
\end{enumerate}
\hfill \mbox{\textit{OCR MEI S1 2010 Q4 [5]}}