| Exam Board | CAIE |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2010 |
| 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 formulas. Given the mean, students find p = 1/4, then apply standard formulas for P(X=5) and P(X>5), and solve an inequality for part (iii). All parts are routine calculations requiring only recall of geometric distribution properties with no problem-solving insight needed. |
| Spec | 5.02g Geometric probabilities: P(X=r) = p(1-p)^(r-1)5.02h Geometric: mean 1/p and variance (1-p)/p^2 |
| Answer | Marks |
|---|---|
| 7 | (i) Find or imply value of p: p = ¼ or 0⋅25 B1 |
| Answer | Marks |
|---|---|
| N > 26⋅4, Nmin = 27 M1 A1 | 3 |
| Answer | Marks | Guidance |
|---|---|---|
| 2 | [7] | |
| Page 5 | Mark Scheme: Teachers’ version | Syllabus |
| GCE A LEVEL – October/November 2010 | 9231 | 02 |
Question 7:
7 | (i) Find or imply value of p: p = ¼ or 0⋅25 B1
Find P(X = 5): (1 – p) 4 p or q 4 p = 0⋅0791 M1 A1
(ii) Find P(X ≥ 5): 1 – (1 + q + q 2 + q 3 ) p or q 4 or
q 4 p + q 5 = 0⋅316 M1 A1
(iii) Find least N with P(X ≤ N) > 0⋅9995: 1 – q N > 0⋅9995, q N < 0⋅0005
N > 26⋅4, Nmin = 27 M1 A1 | 3
2
2 | [7]
Page 5 | Mark Scheme: Teachers’ version | Syllabus | Paper
GCE A LEVEL – October/November 2010 | 9231 | 02The discrete random variable $X$ has a geometric distribution with mean 4.
Find
\begin{enumerate}[label=(\roman*)]
\item P$(X = 5)$, [3]
\item P$(X > 5)$, [2]
\item the least integer $N$ such that P$(X \leqslant N) > 0.9995$. [2]
\end{enumerate}
\hfill \mbox{\textit{CAIE FP2 2010 Q7 [7]}}