| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2013 |
| Session | November |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Poisson distribution |
| Type | Finding maximum n for P(X=0) threshold |
| Difficulty | Standard +0.8 Part (i) requires scaling the Poisson parameter from 150 minutes to 10 hours (600 minutes) and computing a cumulative probability—straightforward but multi-step. Part (ii) is more challenging: students must set up P(X=0)=e^{-λt}≥0.99, solve the inequality involving logarithms, and convert units correctly. This inverse problem with inequality constraint and unit conversion elevates it above routine exercises. |
| 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 |
|---|---|---|
| \(\lambda = 2.8\) | B1 | seen |
| \(e^{-2.8}(1 + 2.8 + \frac{2.8^2}{2})\) | M1 | any \(\lambda\) allow one end error |
| \(= 0.469\) (3 s.f.) or \(0.47(0)\) | A1 [3] | As final answer |
| Answer | Marks | Guidance |
|---|---|---|
| \(e^{-0.7n} \geq 0.99\) | M1 | Allow '=' throughout |
| or \(e^{\lambda} \geq 0.99\) | M1 | Attempt ln both sides |
| \(-0.7n \geq \ln(0.99)\) or \(-\lambda \geq \ln 0.99\) | ||
| \(n \leq 0.01436\) or \(\lambda \leq 0.01005\) | A1 | Can be implied. Accept 3 s.f. |
| '0.01436' \(\times 150\) or '0.01005' \(\times 150 = 0.7\) | M1 | Note \(e^{-(0.7/150)n} \geq 0.99\) scores 1st and 3rd M1 |
| Max period is 2.15 mins (3 sf) | A1 [5] | T & I leading to ans 2.2 mins, SC: B2 |
**(i)**
$\lambda = 2.8$ | B1 | seen
$e^{-2.8}(1 + 2.8 + \frac{2.8^2}{2})$ | M1 | any $\lambda$ allow one end error
$= 0.469$ (3 s.f.) or $0.47(0)$ | A1 [3] | As final answer
**(ii)**
$e^{-0.7n} \geq 0.99$ | M1 | Allow '=' throughout
or $e^{\lambda} \geq 0.99$ | M1 | Attempt ln both sides
$-0.7n \geq \ln(0.99)$ or $-\lambda \geq \ln 0.99$ | |
$n \leq 0.01436$ or $\lambda \leq 0.01005$ | A1 | Can be implied. Accept 3 s.f.
'0.01436' $\times 150$ or '0.01005' $\times 150 = 0.7$ | M1 | Note $e^{-(0.7/150)n} \geq 0.99$ scores 1st and 3rd M1
Max period is 2.15 mins (3 sf) | A1 [5] | T & I leading to ans 2.2 mins, SC: B24 The number of radioactive particles emitted per 150-minute period by some material has a Poisson distribution with mean 0.7.\\
(i) Find the probability that at most 2 particles will be emitted during a randomly chosen 10 -hour period.\\
(ii) Find, in minutes, the longest time period for which the probability that no particles are emitted is at least 0.99 .
\hfill \mbox{\textit{CAIE S2 2013 Q4 [8]}}