| Exam Board | OCR |
|---|---|
| Module | S2 (Statistics 2) |
| Session | Specimen |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Approximating Binomial to Normal Distribution |
| Type | Single probability inequality |
| Difficulty | Moderate -0.3 This is a straightforward application of the normal approximation to the binomial distribution. Part (i) is trivial probability multiplication (1/36), and part (ii) requires standard technique: identify binomial parameters, check approximation validity, apply continuity correction, and use normal tables. The question is slightly easier than average because it's a direct textbook-style application with clear setup and no conceptual obstacles. |
| Spec | 2.03a Mutually exclusive and independent events2.04d Normal approximation to binomial5.04a Linear combinations: E(aX+bY), Var(aX+bY) |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(\frac{1}{36}\) | B1 | For correct probability |
| (ii) Number obtaining two sixes ~ \(\text{B}(60, \frac{1}{36})\) | M1 | For stating or implying binomial distribution |
| Approximate distribution is \(\text{Po}(\frac{5}{3})\) | A1 | For the correct Poisson approximation |
| \(\text{P}(\geq 4) = 1 - e^{-1}\left\{1 + \frac{5}{3} + \frac{(5/3)^2}{2!} + \frac{(5/3)^3}{3!}\right\}\) | M1 | For calculation of correct terms |
| M1 | For correct use of Poisson formula | |
| \(= 0.0883\) | A1 | For correct answer 0.088(3) |
(i) $\frac{1}{36}$ | B1 | For correct probability
(ii) Number obtaining two sixes ~ $\text{B}(60, \frac{1}{36})$ | M1 | For stating or implying binomial distribution
Approximate distribution is $\text{Po}(\frac{5}{3})$ | A1 | For the correct Poisson approximation
$\text{P}(\geq 4) = 1 - e^{-1}\left\{1 + \frac{5}{3} + \frac{(5/3)^2}{2!} + \frac{(5/3)^3}{3!}\right\}$ | M1 | For calculation of correct terms
| M1 | For correct use of Poisson formula
$= 0.0883$ | A1 | For correct answer 0.088(3)
**Total: 6 marks**
---3 Sixty people each make two throws with a fair six-sided die.\\
(i) State the probability of one particular person obtaining two sixes.\\
(ii) Using a suitable approximation, calculate the probability that at least four of the sixty obtain two sixes.
\hfill \mbox{\textit{OCR S2 Q3 [6]}}