| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2022 |
| Session | November |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Approximating the Binomial to the Poisson distribution |
| Type | Calculate multiple probabilities using Poisson approximation |
| Difficulty | Moderate -0.8 This is a straightforward application of the Poisson approximation to the binomial distribution. Students need to recognize n is large (200) and p is small (0.016), calculate λ = np = 3.2, then find P(X > 3) using tables or calculator. The justification in part (b) simply requires stating the standard conditions (large n, small p). This is a routine textbook exercise with no problem-solving insight required, making it easier than average. |
| 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 |
|---|---|---|
| 3(a) | Use of Poisson. mean = 3.2 | B1 B1 |
| Answer | Marks | Guidance |
|---|---|---|
| or 1 – (0.04076 + 0.1304 + 0.2087 + 0.2226) | M1 | Allow any λ . |
| Answer | Marks | Guidance |
|---|---|---|
| = 0.397 or 0.398 | A1 | SC Use of binomial: B1 for answer 0.398 (3 sf). |
| Answer | Marks | Guidance |
|---|---|---|
| Question | Answer | Marks |
| Answer | Marks | Guidance |
|---|---|---|
| 3(b) | [Binomial with] [n =] 200 > 50 | B1 |
| [np =][200 × 0.016 =] 3.2 < 5 or [p =]0.016 < 0.1 | B1 | If B0 B0 |
| Answer | Marks | Guidance |
|---|---|---|
| Question | Answer | Marks |
Question 3:
--- 3(a) ---
3(a) | Use of Poisson. mean = 3.2 | B1 B1
3.22 3.23
1−e−3.2 1+3.2+ + or 1 – e–3.2 (1 + 3.2 + 5.12 + 5.46133)
2 3!
or 1 – (0.04076 + 0.1304 + 0.2087 + 0.2226) | M1 | Allow any λ .
Allow one end error.
= 0.397 or 0.398 | A1 | SC Use of binomial: B1 for answer 0.398 (3 sf).
0.397 or 0.398 with no working scores SC B1.
4
Question | Answer | Marks | Guidance
--- 3(b) ---
3(b) | [Binomial with] [n =] 200 > 50 | B1
[np =][200 × 0.016 =] 3.2 < 5 or [p =]0.016 < 0.1 | B1 | If B0 B0
SC n large (or n > 50),
and p small or p < 0.1 or np <5 : B1.
2
Question | Answer | Marks | Guidance1.6% of adults in a certain town ride a bicycle. A random sample of 200 adults from this town is selected.
\begin{enumerate}[label=(\alph*)]
\item Use a suitable approximating distribution to find the probability that more than 3 of these adults ride a bicycle. [4]
\item Justify your approximating distribution. [2]
\end{enumerate}
\hfill \mbox{\textit{CAIE S2 2022 Q3 [6]}}