| Exam Board | OCR |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2012 |
| Session | January |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Binomial Distribution |
| Type | Identify distribution and parameters |
| Difficulty | Moderate -0.8 This is a straightforward S1 question requiring identification of a binomial distribution from context (likely given in the problem setup), stating its parameters n and p, listing standard conditions (fixed n, constant p, independence), then performing basic binomial probability calculations using either tables or formula. All steps are routine recall and standard procedure with no problem-solving insight required. |
| Spec | 5.02f Geometric distribution: conditions5.02g Geometric probabilities: P(X=r) = p(1-p)^(r-1)5.02h Geometric: mean 1/p and variance (1-p)/p^2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\text{Geo}(0.6)\) or \(G(0.6)\) or Geo with \(p = 0.6\) | B1B1 | Allow \(\text{Geo}(60\%)\) B1B1; \(\text{Bin}(\ldots, 0.6)\) B0B1 |
| \(P(\text{woman})\) const or chance of woman const; Each voter has same prob | B1 | or %age of women is constant; In context |
| Whether one voter is a woman is indep of whether any other is a woman | B1 | Allow: "voter(s) independent", "Men & women are independent", "\(P(\text{woman})\) is indep", "Each woman is indep". EACH comment must be in context |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(0.4^3 \times 0.6\) | M1 | ft their \(\text{Geo}(p)\) from (i) |
| \(= \frac{24}{625}\) or \(0.0384\) | A1f | ft their \(\text{Geo}(p)\) from (i). Allow \(0.3^3 \times 0.6\) but no other \(q^3 \times 0.6\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(0.4^3\) alone, or \((0.4^4 + 0.4^3{\times}0.6)\) or \((0.4^4 + \text{(ii)})\) | M1 | \(1-(0.6 + 0.4{\times}0.6 + 0.4^2{\times}0.6)\) (allow extra term \(0.4^3{\times}0.6\)); ft their \(\text{Geo}(p)\) from (i) |
| \(= \frac{8}{125}\) or \(0.064\) | A1f | ft their \(\text{Geo}(p)\) from (i) |
## Question 7:
### Part (i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\text{Geo}(0.6)$ or $G(0.6)$ or Geo with $p = 0.6$ | B1B1 | Allow $\text{Geo}(60\%)$ B1B1; $\text{Bin}(\ldots, 0.6)$ B0B1 |
| $P(\text{woman})$ const or chance of woman const; Each voter has same prob | B1 | or %age of women is constant; In context |
| Whether one voter is a woman is indep of whether any other is a woman | B1 | Allow: "voter(s) independent", "Men & women are independent", "$P(\text{woman})$ is indep", "Each woman is indep". EACH comment must be in context |
### Part (ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $0.4^3 \times 0.6$ | M1 | ft their $\text{Geo}(p)$ from (i) |
| $= \frac{24}{625}$ or $0.0384$ | A1f | ft their $\text{Geo}(p)$ from (i). Allow $0.3^3 \times 0.6$ but no other $q^3 \times 0.6$ |
### Part (iii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $0.4^3$ alone, or $(0.4^4 + 0.4^3{\times}0.6)$ or $(0.4^4 + \text{(ii)})$ | M1 | $1-(0.6 + 0.4{\times}0.6 + 0.4^2{\times}0.6)$ (allow extra term $0.4^3{\times}0.6$); ft their $\text{Geo}(p)$ from (i) |
| $= \frac{8}{125}$ or $0.064$ | A1f | ft their $\text{Geo}(p)$ from (i) |
---(i) State a suitable distribution that can be used as a model for $X$, giving the value(s) of any parameter(s). State also any necessary condition(s) for this distribution to be a good model.
Use the distribution stated in part (i) to find\\
(ii) $\mathrm { P } ( X = 4 )$,\\
(iii) $\mathrm { P } ( X \geqslant 4 )$.
\hfill \mbox{\textit{OCR S1 2012 Q7 [8]}}