| Exam Board | CAIE |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2012 |
| Session | November |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Geometric Distribution |
| Type | Find minimum n for P(X ≤ n) > threshold |
| Difficulty | Standard +0.3 This is a straightforward application of geometric distribution with standard formulas: P(X=n) = (1-p)^(n-1) × p for part (i), summing for part (ii), and solving P(X≤n) ≥ 0.95 using logarithms for part (iii). While part (iii) requires solving an inequality with logs, this is a routine technique for geometric distribution questions at Further Maths level, making it slightly above average difficulty overall. |
| Spec | 5.02f Geometric distribution: conditions5.02g Geometric probabilities: P(X=r) = p(1-p)^(r-1) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \((1-0.2)^{19} \times 0.2 = 0.00288\) | M1 A1 | Find prob. that first snow falls on \(20^{\text{th}}\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(1 - (1-0.2)^4 = 0.59[0]\) | M1 A1 | Find prob. that first snow falls before \(5^{\text{th}}\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(1-(1-0.2)^n \geq 0.95,\quad 0.8^n \leq 0.05\) | M1 | Formulate condition for day \(n\) of month |
| \(n > \frac{\log 0.05}{\log 0.8}\) | M1 | Take logs (any base) to give bound for \(n\) |
| \(n > 13.4,\quad n_{\min} = 14\) | A1 | |
| Subtotal: 3 | Total: [7] |
## Question 6(i):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $(1-0.2)^{19} \times 0.2 = 0.00288$ | M1 A1 | Find prob. that first snow falls on $20^{\text{th}}$ |
**Subtotal: 2**
### Question 6(ii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $1 - (1-0.2)^4 = 0.59[0]$ | M1 A1 | Find prob. that first snow falls before $5^{\text{th}}$ |
**Subtotal: 2**
### Question 6(iii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $1-(1-0.2)^n \geq 0.95,\quad 0.8^n \leq 0.05$ | M1 | Formulate condition for day $n$ of month |
| $n > \frac{\log 0.05}{\log 0.8}$ | M1 | Take logs (any base) to give bound for $n$ |
| $n > 13.4,\quad n_{\min} = 14$ | A1 | |
**Subtotal: 3 | Total: [7]**
---6 In a skiing resort, for each day during the winter season, the probability that snow will fall on that day is 0.2 , independently of any other day. The first day of the winter season is 1 December. Find, for the winter season,\\
(i) the probability that the first snow falls on 20 December,\\
(ii) the probability that the first snow falls before 5 December,\\
(iii) the earliest date in December such that the probability that the first snow falls on or before that date is at least 0.95 .
\hfill \mbox{\textit{CAIE FP2 2012 Q6 [7]}}