| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2020 |
| Session | Specimen |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Poisson distribution |
| Type | Single time period probability |
| Difficulty | Moderate -0.5 This appears to be a standard Poisson distribution question asking about probability in a single time period with a given mean. Such questions typically involve direct application of the Poisson formula P(X=r) = e^(-λ)λ^r/r!, which is routine for S2 level with minimal problem-solving required beyond formula substitution and calculator work. |
| Spec | 5.02i Poisson distribution: random events model |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(e^{-3.84} \times \dfrac{3.84^4}{4!}\) | 1, M1 | |
| \(= 0.195\) (3 sf) | 1, A1 | |
| Total: 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(1 - P(X = 0, 1, 2)\) | 1, M1 | Attempted |
| \(1 - e^{-1.44}\left(1 + 1.44 + \dfrac{1.44^2}{2}\right)\) | 1, M1 | |
| \(= 0.176\) | 1, A1 | |
| Total: 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(X \sim N(41, 41)\) | 1, B1 | Seen or implied |
| \(\dfrac{40.5 - 41}{\sqrt{41}}\) \((= -0.078)\) \(\dfrac{59.5 - 41}{\sqrt{41}}\) \((= 2.889)\) | 2, M1M1 | M1 for each or M1M0 if no continuity correction (cc) or square root sign, or incorrect cc in both |
| \(\Phi(2.889') - \Phi(-0.078')\) \(= \Phi(2.889') - (1 - \Phi(0.078'))\) \(= 0.9981 - (1 - 0.5311)\) | 1, M1 | Use of tables and correct area consistent with their working |
| \(= 0.529\) (3 sf) | 1, A1 | Correct working only |
| Total: 5 |
## Question 3(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $e^{-3.84} \times \dfrac{3.84^4}{4!}$ | 1, M1 | |
| $= 0.195$ (3 sf) | 1, A1 | |
| **Total: 2** | | |
---
## Question 3(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $1 - P(X = 0, 1, 2)$ | 1, M1 | Attempted |
| $1 - e^{-1.44}\left(1 + 1.44 + \dfrac{1.44^2}{2}\right)$ | 1, M1 | |
| $= 0.176$ | 1, A1 | |
| **Total: 3** | | |
---
## Question 3(c):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $X \sim N(41, 41)$ | 1, B1 | Seen or implied |
| $\dfrac{40.5 - 41}{\sqrt{41}}$ $(= -0.078)$ $\dfrac{59.5 - 41}{\sqrt{41}}$ $(= 2.889)$ | 2, M1M1 | M1 for each or M1M0 if no continuity correction (cc) or square root sign, or incorrect cc in both |
| $\Phi(2.889') - \Phi(-0.078')$ $= \Phi(2.889') - (1 - \Phi(0.078'))$ $= 0.9981 - (1 - 0.5311)$ | 1, M1 | Use of tables and correct area consistent with their working |
| $= 0.529$ (3 sf) | 1, A1 | Correct working only |
| **Total: 5** | | |
---3 The number of calls received at a small call centre has a Poisson distribution with mean 2.4 calls per 5-minute period.\\
(a) Find the probability of exactly 4 calls in an 8 -minute period.\\
(b) Find the probability of at least 3 calls in a 3-minute period.\\
The number of calls received at a large call centre has a Poisson distribution with mean 41 calls per 5-minute period.\\
(c) Use an approximating distribution to find the probability that the number of calls received in a 5 -minute period is between 41 and 59 inclusive.\\
\hfill \mbox{\textit{CAIE S2 2020 Q3 [10]}}