| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2020 |
| Session | March |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Poisson distribution |
| Type | Finding maximum n for P(X=0) threshold |
| Difficulty | Standard +0.3 Part (a) is a straightforward Poisson calculation requiring scaling the mean from 50 days to 365 days, then computing P(X<3). Part (b) requires solving P(X=0)=e^{-λn}>0.95 for n, involving logarithms and rate conversion—slightly more challenging but still a standard textbook exercise with clear methodology. |
| Spec | 5.02i Poisson distribution: random events model5.02j Poisson formula: P(X=x) = e^(-lambda)*lambda^x/x! |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(\lambda = 0.4 \times 365 \div 50 = 2.92\) | B1 | |
| \(e^{-2.92}(1 + 2.92 + \frac{2.92^2}{2})\) | M1 | Any \(\lambda\). Allow one end error |
| \(= 0.441\) (3 sf) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(e^{-\lambda} > 0.95\) | M1 | Allow '\(=\)' throughout |
| \(-\lambda > \ln 0.95\) or \(\lambda < 0.051293\) OE | M1 | Attempt ln both sides |
| \('0.051293' \times 50 \div 0.4 \; (= 6.411)\) | M1 | |
| Largest \(n\) is 6 (3 sf). Allow \(n = 6\) or \(n \leqslant 6\) (NOT \(n < 6\) or \(n \geqslant 6\) as final answer) | A1 | SC Trial and Improvement: M1 for \(e^{-\lambda} > 0.95\) SOI; M1 for \(\lambda = n \times \frac{0.4}{50}\); M1 for use of both; \(n=6\) giving 0.9531 and \(n=7\) giving 0.9455; A1 \(n=6\) |
## Question 4(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| $\lambda = 0.4 \times 365 \div 50 = 2.92$ | B1 | |
| $e^{-2.92}(1 + 2.92 + \frac{2.92^2}{2})$ | M1 | Any $\lambda$. Allow one end error |
| $= 0.441$ (3 sf) | A1 | |
## Question 4(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| $e^{-\lambda} > 0.95$ | M1 | Allow '$=$' throughout |
| $-\lambda > \ln 0.95$ or $\lambda < 0.051293$ OE | M1 | Attempt ln both sides |
| $'0.051293' \times 50 \div 0.4 \; (= 6.411)$ | M1 | |
| Largest $n$ is 6 (3 sf). Allow $n = 6$ or $n \leqslant 6$ (NOT $n < 6$ or $n \geqslant 6$ as final answer) | A1 | SC Trial and Improvement: M1 for $e^{-\lambda} > 0.95$ SOI; M1 for $\lambda = n \times \frac{0.4}{50}$; M1 for use of both; $n=6$ giving 0.9531 and $n=7$ giving 0.9455; A1 $n=6$ |4 The number of accidents on a certain road has a Poisson distribution with mean 0.4 per 50-day period.
\begin{enumerate}[label=(\alph*)]
\item Find the probability that there will be fewer than 3 accidents during a year (365 days).
\item The probability that there will be no accidents during a period of $n$ days is greater than 0.95 . Find the largest possible value of $n$.
\end{enumerate}
\hfill \mbox{\textit{CAIE S2 2020 Q4 [7]}}