| Exam Board | CAIE |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2010 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Exponential Distribution |
| Type | Multiple independent components |
| Difficulty | Standard +0.8 This is a Further Maths question requiring understanding of exponential distributions, independence, and probability calculations. Part (a) involves standard applications with multiple components (requiring binomial probability), while part (b) requires algebraic proof connecting individual and collective survival probabilities—a conceptual step beyond routine calculation that demonstrates the memoryless property. |
| Spec | 5.02i Poisson distribution: random events model5.02j Poisson formula: P(X=x) = e^(-lambda)*lambda^x/x!5.02k Calculate Poisson probabilities |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Find prob. of 5 operating after 3 months: \(\{\exp(-3/2.5)\}^5 = \{\exp(-3\times 0.4)\}^5\) | ||
| \(= 0.3011^5 = 0.00248\) (to 2 sf) | M1 A1 | Subtotal: 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Find prob. of 1 failing within one month: \(p_1 = 1 - \exp(-1/2.5)\) \([= 0.3297]\) | B1 | |
| Find prob. of 2 failing within one month: \(^5C_2\, p_1^2(1-p_1)^3 = 0.327\) | M1 A1 | Using 2.5 as parameter, not mean, can earn M1s only in (a). Subtotal: 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Find prob. of \(n\) operating after \(c\) months: \(\{\exp(-c\lambda)\}^n\) \([\lambda = 0.4]\) | M1 A1 | |
| Show same as 1 operating after \(nc\) months: \(= \exp(-nc\lambda)\) | M1 A1 | Total: 9 |
## Question 8:
### Part (a)(i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Find prob. of 5 operating after 3 months: $\{\exp(-3/2.5)\}^5 = \{\exp(-3\times 0.4)\}^5$ | | |
| $= 0.3011^5 = 0.00248$ (to 2 sf) | M1 A1 | **Subtotal: 2** |
### Part (a)(ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Find prob. of 1 failing within one month: $p_1 = 1 - \exp(-1/2.5)$ $[= 0.3297]$ | B1 | |
| Find prob. of 2 failing within one month: $^5C_2\, p_1^2(1-p_1)^3 = 0.327$ | M1 A1 | Using 2.5 as parameter, not mean, can earn M1s only in (a). **Subtotal: 3** |
### Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Find prob. of $n$ operating after $c$ months: $\{\exp(-c\lambda)\}^n$ $[\lambda = 0.4]$ | M1 A1 | |
| Show same as 1 operating after $nc$ months: $= \exp(-nc\lambda)$ | M1 A1 | **Total: 9** |
---8 A certain mechanical component has a lifetime, $T$ months, which has a negative exponential distribution with mean 2.5.
\begin{enumerate}[label=(\alph*)]
\item A machine is fitted with 5 of these components which function independently.
\begin{enumerate}[label=(\roman*)]
\item Find the probability that all 5 components are operating 3 months after being fitted.
\item Find also the probability that exactly two components fail within one month of being fitted.
\end{enumerate}\item Show that the probability that $n$ independent components are all operating $c$ months after being fitted is equal to the probability that a single component is operating $n c$ months after being fitted.
\end{enumerate}
\hfill \mbox{\textit{CAIE FP2 2010 Q8 [9]}}