| Exam Board | CAIE |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2019 |
| Session | November |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Exponential Distribution |
| Type | Find threshold for given probability |
| Difficulty | Standard +0.3 This is a straightforward application of the exponential distribution with standard bookwork (mean = 1/a) and routine probability calculations. Part (iii) requires setting up an inequality with complements and logarithms, but follows a predictable pattern for 'at least one' problems. Slightly easier than average due to the direct application of formulas with minimal problem-solving insight required. |
| Spec | 5.03a Continuous random variables: pdf and cdf5.03b Solve problems: using pdf |
| Answer | Marks | Guidance |
|---|---|---|
| 7(i) | a = 1/200 or 0⋅005 | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| 7(ii) | p = P(T < 150) = F(150) = 1 – e-150a | M1 |
| p = 1 – e - 0⋅75 = 0⋅528 | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| 7(iii) | 1 – p n > 0⋅99 | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| or 0⋅01 > 0⋅528 n | A1 | |
| n > log 0⋅01 / log 0⋅528 | M1 | Rearrange and take logs to give bound |
| Answer | Marks | Guidance |
|---|---|---|
| min | A1 | Find n |
| Answer | Marks | Guidance |
|---|---|---|
| Question | Answer | Marks |
Question 7:
--- 7(i) ---
7(i) | a = 1/200 or 0⋅005 | B1 | State a or find a by equating mean value to 1/a
1
--- 7(ii) ---
7(ii) | p = P(T < 150) = F(150) = 1 – e-150a | M1 | Find P(T < 150)
p = 1 – e - 0⋅75 = 0⋅528 | A1
2
--- 7(iii) ---
7(iii) | 1 – p n > 0⋅99 | M1 | Formulate condition for n
0⋅01 > (1 – e - 0⋅75) n
or 0⋅01 > 0⋅528 n | A1
n > log 0⋅01 / log 0⋅528 | M1 | Rearrange and take logs to give bound
n > 7⋅20 [or 7⋅21] so n = 8
min | A1 | Find n
min
4
Question | Answer | Marks | GuidanceThe time, $T$ days, before an electrical component develops a fault has distribution function F given by
$$\mathrm{F}(t) = \begin{cases} 1 - e^{-at} & t \geqslant 0, \\ 0 & \text{otherwise}, \end{cases}$$
where $a$ is a positive constant. The mean value of $T$ is 200.
\begin{enumerate}[label=(\roman*)]
\item Write down the value of $a$.
[1]
\item Find the probability that an electrical component of this type develops a fault in less than 150 days.
[2]
\end{enumerate}
A piece of equipment contains $n$ of these components, which develop faults independently of each other. The probability that, after 150 days, at least one of the $n$ components has not developed a fault is greater than 0.99.
\begin{enumerate}[label=(\roman*)]
\setcounter{enumi}{2}
\item Find the smallest possible value of $n$.
[4]
\end{enumerate}
\hfill \mbox{\textit{CAIE FP2 2019 Q7 [7]}}