| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2013 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Approximating the Binomial to the Poisson distribution |
| Type | State Poisson approximation with justification |
| Difficulty | Moderate -0.8 This is a straightforward application of the Poisson approximation to the binomial distribution. Students need to identify B(400, 0.012), recognize that n is large and p is small (np = 4.8), state Po(4.8) as the approximation, and perform a basic Poisson probability calculation. All steps are routine and follow standard textbook procedures with no novel problem-solving required. |
| Spec | 5.02b Expectation and variance: discrete random variables5.02c Linear coding: effects on mean and variance5.02i Poisson distribution: random events model5.02j Poisson formula: P(X=x) = e^(-lambda)*lambda^x/x!5.02k Calculate Poisson probabilities5.02l Poisson conditions: for modelling |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Binomial | B1 | |
| \(n = 400\), \(p = 0.012\) | B1 [2] | Both. Not \(p = 1.2\%\). Or B(400, 0.012): B1B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Poisson | B1 | |
| \(n\) large and mean \(= 4.8\), which is \(< 5\) | B1 [2] | \(n\) large, \(p\) small |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(1 - e^{-4.8}(1 + 4.8 + \frac{4.8^2}{2})\) | M1 | \(P(X = 0, 1, 2)\); allow any \(\lambda\); allow one end error |
| \(= 0.857/0.858\) | A1 [2] | (Normal/Binomial in (ii) can score M1 only) |
## Question 1:
### Part (i)
| Answer/Working | Marks | Guidance |
|---|---|---|
| Binomial | B1 | |
| $n = 400$, $p = 0.012$ | B1 [2] | Both. Not $p = 1.2\%$. Or B(400, 0.012): B1B1 |
### Part (ii)
| Answer/Working | Marks | Guidance |
|---|---|---|
| Poisson | B1 | |
| $n$ large and mean $= 4.8$, which is $< 5$ | B1 [2] | $n$ large, $p$ small |
### Part (iii)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $1 - e^{-4.8}(1 + 4.8 + \frac{4.8^2}{2})$ | M1 | $P(X = 0, 1, 2)$; allow any $\lambda$; allow one end error |
| $= 0.857/0.858$ | A1 [2] | (Normal/Binomial in (ii) can score M1 only) |
---1 It is known that $1.2 \%$ of rods made by a certain machine are bent. The random variable $X$ denotes the number of bent rods in a random sample of 400 rods.\\
(i) State the distribution of $X$.\\
(ii) State, with a reason, a suitable approximate distribution for $X$.\\
(iii) Use your approximate distribution to find the probability that the sample will include more than 2 bent rods.
\hfill \mbox{\textit{CAIE S2 2013 Q1 [6]}}