| Exam Board | CAIE |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2019 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Geometric Distribution |
| Type | Find maximum n for P(X > n) > threshold |
| Difficulty | Standard +0.3 This is a straightforward geometric distribution question with standard calculations. Parts (i)-(iii) involve direct formula application (mean = 1/p, P(X=6), P(X>4)), while part (iv) requires solving an inequality involving logarithms—a routine Further Maths technique. The setup is clear with p=1/3, and all parts follow textbook methods without requiring novel insight or complex multi-step reasoning. |
| Spec | 5.02g Geometric probabilities: P(X=r) = p(1-p)^(r-1) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Mean \(= 3\) | B1 | State mean of \(X\) |
| 1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(P(X=6) = q^5p\) with \(p = \frac{1}{3}\), \(q = \frac{2}{3}\) \(= \frac{32}{729}\) or \(0.0439\) | B1 | Find probability of score of 3 or 4 on exactly 6 throws |
| 1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(P(X > 4) = q^4 = \frac{16}{81}\) or \(0.1975\) or \(0.198\) | M1 A1 | Find probability of score of 3 or 4 on more than 4 throws |
| 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(1 - q^{n-1} < 0.95\) | M1 | Formulate condition for \(n\) (\(1 - q^n\) is M0) |
| \(0.05 < (\frac{2}{3})^{n-1}\), \(n - 1 < \frac{\log 0.05}{\log \frac{2}{3}}\) | M1 | Set \(q = \frac{2}{3}\), rearrange and take logs (any base) to give bound |
| \(n - 1 < 7.39\), \(n_{max} = 8\) | A1 | Find \(n_{max}\) (\(>\) or \(=\) can earn M1 M1 A0, max \(\frac{2}{3}\)) |
| 3 |
## Question 6(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Mean $= 3$ | B1 | State mean of $X$ |
| | **1** | |
## Question 6(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $P(X=6) = q^5p$ with $p = \frac{1}{3}$, $q = \frac{2}{3}$ $= \frac{32}{729}$ or $0.0439$ | B1 | Find probability of score of 3 or 4 on exactly 6 throws |
| | **1** | |
## Question 6(iii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $P(X > 4) = q^4 = \frac{16}{81}$ or $0.1975$ or $0.198$ | M1 A1 | Find probability of score of 3 or 4 on more than 4 throws |
| | **2** | |
## Question 6(iv):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $1 - q^{n-1} < 0.95$ | M1 | Formulate condition for $n$ ($1 - q^n$ is M0) |
| $0.05 < (\frac{2}{3})^{n-1}$, $n - 1 < \frac{\log 0.05}{\log \frac{2}{3}}$ | M1 | Set $q = \frac{2}{3}$, rearrange and take logs (any base) to give bound |
| $n - 1 < 7.39$, $n_{max} = 8$ | A1 | Find $n_{max}$ ($>$ or $=$ can earn M1 M1 A0, max $\frac{2}{3}$) |
| | **3** | |6 A fair six-sided die is thrown until a 3 or a 4 is obtained. The number of throws taken is denoted by the random variable $X$.\\
(i) State the mean value of $X$.\\
(ii) Find the probability that obtaining a 3 or a 4 takes exactly 6 throws.\\
(iii) Find the probability that obtaining a 3 or a 4 takes more than 4 throws.\\
(iv) Find the greatest integer $n$ such that the probability of obtaining a 3 or a 4 in fewer than $n$ throws is less than 0.95.\\
\hfill \mbox{\textit{CAIE FP2 2019 Q6 [7]}}