| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2022 |
| Session | November |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Poisson distribution |
| Type | Single period normal approximation - scaled period (normal approximation only) |
| Difficulty | Standard +0.3 Part (a) requires adjusting the Poisson rate for a different time period and calculating P(X ≥ 3), which is straightforward. Part (b) involves recognizing when to use normal approximation to Poisson (large λ) and applying continuity correction—standard A-level technique but requires understanding of when approximations are appropriate. The calculations are routine once the setup is correct. |
| Spec | 5.02i Poisson distribution: random events model5.02k Calculate Poisson probabilities5.04a Linear combinations: E(aX+bY), Var(aX+bY) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\lambda = 5.2 \div 2 \quad [= 2.6]\) | B1 | |
| \(1 - e^{-2.6}(1 + 2.6 + \frac{2.6^2}{2})\) or \(1 - e^{-2.6}(1 + 2.6 + 3.38)\) or \(1-(0.07427 + 0.1931 + 0.2510)\) | M1 | Allow any \(\lambda\). Allow one end error. Must see expression. |
| \(= 0.482\) (3 sf) | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(N(120 \times 5.2,\ 120 \times 5.2)\) | B1 | Stated or implied. Give at early stage. |
| \(\dfrac{649.5 - \textit{their } `624`}{\sqrt{\textit{their } `624`}} \quad [= 1.021]\) | M1 | Allow with no or wrong continuity correction. |
| \(1 - \Phi(\textit{their } `1.021`)\) | M1 | For area consistent with *their* working. |
| \(= 0.154\) (3 sf) | A1 |
## Question 3(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\lambda = 5.2 \div 2 \quad [= 2.6]$ | B1 | |
| $1 - e^{-2.6}(1 + 2.6 + \frac{2.6^2}{2})$ or $1 - e^{-2.6}(1 + 2.6 + 3.38)$ **or** $1-(0.07427 + 0.1931 + 0.2510)$ | M1 | Allow any $\lambda$. Allow one end error. Must see expression. |
| $= 0.482$ (3 sf) | B1 | |
---
## Question 3(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $N(120 \times 5.2,\ 120 \times 5.2)$ | B1 | Stated or implied. Give at early stage. |
| $\dfrac{649.5 - \textit{their } `624`}{\sqrt{\textit{their } `624`}} \quad [= 1.021]$ | M1 | Allow with no or wrong continuity correction. |
| $1 - \Phi(\textit{their } `1.021`)$ | M1 | For area consistent with *their* working. |
| $= 0.154$ (3 sf) | A1 | |
---3 Drops of water fall randomly from a leaking tap at a constant average rate of 5.2 per minute.
\begin{enumerate}[label=(\alph*)]
\item Find the probability that at least 3 drops fall during a randomly chosen 30 -second period.
\item Use a suitable approximating distribution to find the probability that at least 650 drops fall during a randomly chosen 2-hour period.
\end{enumerate}
\hfill \mbox{\textit{CAIE S2 2022 Q3 [7]}}