| Exam Board | OCR MEI |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2013 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Binomial Distribution |
| Type | Nested binomial expected count |
| Difficulty | Standard +0.3 This is a straightforward nested binomial problem requiring three standard calculations: (i) direct binomial probability P(X=5) with n=50, p=0.1; (ii) complement probability 1-P(X=0) with n=20, p=0.1; (iii) expectation E=np using the probability from (ii). All steps are routine applications of binomial distribution formulas with no conceptual challenges beyond recognizing the nested structure, making it slightly easier than average. |
| Spec | 2.04b Binomial distribution: as model B(n,p)2.04c Calculate binomial probabilities |
| Answer | Marks | Guidance |
|---|---|---|
| Marks: M1 | M1 | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Marks: M1 | A1 | [2] |
| Answer | Marks | Guidance |
|---|---|---|
| Marks: M1 | A1 | [2] |
## Part (i)
**Answer:**
$$X \sim B(50, 0.1)$$
$$P(\text{5 underweight}) = \binom{50}{5} \times 0.1^5 \times 0.9^{45} = 0.1849$$
**Marks:** M1 | M1 | A1 | [3]
**Guidance:**
- M1: For $0.1^5 \times 0.9^{45}$
- M1: For $\binom{50}{5} \times p^5 \times q^{45}$. Also for $2118760 \times 8.73 \times 10^{-8}$
- A1: CAO. Allow 0.185 or better. NB 0.18 gets A0
## Part (ii)
**Answer:**
$$X \sim B(20, 0.1)$$
$$P(X \geq 1) = 1 - P(X = 0)$$
$$= 1 - 0.1216 = 0.8784$$
**Marks:** M1 | A1 | [2]
**Guidance:**
- M1: For 0.1216
- A1: CAO. Allow M1 for 0.9²⁰. Allow 0.878 or better. See tables at the website http://www.mei.org.uk/files/pdf/formula_book_mf2.pdf
## Part (iii)
**Answer:**
$$E(X) = 48 \times 0.8784 = 42.16 \text{ (= 42.2)}$$
**Marks:** M1 | A1 | [2]
**Guidance:**
- M1: FT their probability from part (ii)
- A1: If any indication of rounding to 42 or 43 or to another integer on FT allow M1A0. SC1 for $48 \times \text{their } p$ giving an integer answer. NB 0.6083 in (ii) leads to 29.20
---3 The weights of bags of a particular brand of flour are quoted as 1.5 kg . In fact, on average $10 \%$ of bags are underweight.\\
(i) Find the probability that, in a random sample of 50 bags, there are exactly 5 bags which are underweight.\\
(ii) Bags are randomly chosen and packed into boxes of 20 . Find the probability that there is at least one underweight bag in a box.\\
(iii) A crate contains 48 boxes. Find the expected number of boxes in the crate which contain at least one underweight bag.
\hfill \mbox{\textit{OCR MEI S1 2013 Q3 [7]}}