| Exam Board | CAIE |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2017 |
| Session | Specimen |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Geometric Distribution |
| Type | Determine p from given mean or variance |
| Difficulty | Standard +0.3 This is a straightforward application of geometric distribution formulas. Part (i) requires knowing mean = 1/p and variance = (1-p)/p² then solving a simple equation. Parts (ii)-(iv) are direct substitutions into standard formulas. While it's a Further Maths topic, the question requires only formula recall and basic algebraic manipulation with no novel problem-solving or insight. |
| 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 |
|---|---|---|
| \(\frac{1}{p} = 2 \times \frac{(1-p)}{p^2},\quad p = \frac{2}{3}\) | M1 A1 | A.G. |
| Answer | Marks |
|---|---|
| \(P(X = 4) = q^3 \times p = \frac{2}{81}\) *or* \(0.0247\) | B1 |
| Answer | Marks |
|---|---|
| \(= 1 - (1 - q^4) = q^4 = \frac{1}{81}\) *or* \(0.0123\) | M1 A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Formulate condition for \(N\): \(1 - q^N > 0.999\), \([(\frac{1}{3})^N < 0.001]\) | M1 | |
| Take logs (any base) to give bound for \(N\): \(N > \log 0.001 / \log\frac{1}{3}\) | M1 | |
| Find \(N_{\min}\): \(N > 6.29\), \(N_{\min} = 7\) | A1 | (\(N < 6.29\) *or* \(N = 6.29\) earns M2 A0) |
## Question 6(i):
Find prob. $p$ of head from mean $= 2 \times$ variance:
$\frac{1}{p} = 2 \times \frac{(1-p)}{p^2},\quad p = \frac{2}{3}$ | M1 A1 | A.G.
**Total: 2 marks**
---
## Question 6(ii):
Find $P(X = 4)$ (denoting $1 - p$ by $q\;[= \frac{1}{3}]$):
$P(X = 4) = q^3 \times p = \frac{2}{81}$ *or* $0.0247$ | B1 |
**Total: 1 mark**
---
## Question 6(iii):
Find or state $P(X > 4)$:
$P(X > 4) = [1 - (1 + q + q^2 + q^3) \times p]$
$= 1 - (1 - q^4) = q^4 = \frac{1}{81}$ *or* $0.0123$ | M1 A1 |
**Total: 2 marks**
---
## Question 6(iv):
Formulate condition for $N$: $1 - q^N > 0.999$, $[(\frac{1}{3})^N < 0.001]$ | M1 |
Take logs (any base) to give bound for $N$: $N > \log 0.001 / \log\frac{1}{3}$ | M1 |
Find $N_{\min}$: $N > 6.29$, $N_{\min} = 7$ | A1 | ($N < 6.29$ *or* $N = 6.29$ earns M2 A0)
**Total: 3 marks**
---6 A biased coin is tossed repeatedly until a head is obtained. The random variable $X$ denotes the number of tosses required for a head to be obtained. The mean of $X$ is equal to twice the variance of $X$.\\
(i) Show that the probability that a head is obtained when the coin is tossed once is $\frac { 2 } { 3 }$.\\
\includegraphics[max width=\textwidth, alt={}, center]{3b311657-f609-4e8d-81e6-b0cbc7a8cbae-11_69_1571_450_328}\\
(ii) Find $\mathrm { P } ( X = 4 )$.\\
(iii) Find $\mathrm { P } ( X > 4 )$.\\
(iv) Find the least integer $N$ such that $\mathrm { P } ( X \leqslant N ) > 0.999$.\\
\hfill \mbox{\textit{CAIE FP2 2017 Q6 [8]}}