| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2019 |
| Session | November |
| Marks | 7 |
| 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 n=500, p=1/150, verify the standard conditions (n large, p small, np moderate), state Po(10/3), and perform a basic probability calculation. All steps are routine and follow a standard textbook template with no novel problem-solving required. |
| Spec | 2.04b Binomial distribution: as model B(n,p)2.04d Normal approximation to binomial5.02i Poisson distribution: random events model5.02k Calculate Poisson probabilities |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Binomial | B1 | |
| \(n = 500\) and \(p = \frac{1}{150}\) or \(0.00667\) | B1 | Or \(B\left(500, \frac{1}{150}\right)\) for B1B1 |
| Total: 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Poisson | B1 | |
| \(n\) large and mean \(= \frac{10}{3}\) or \(3.3\) or better, which is \(< 5\) | B1 | Accept \(n > 50\) |
| Total: 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(1 - e^{-\frac{10}{3}} \times \left(1 + \frac{10}{3} + \frac{\left(\frac{10}{3}\right)^2}{2}\right)\) | M1 | \(1 - P(X = 0, 1, 2)\) |
| \(= 1 - 0.353\) | A1 | Correct expression with \(\lambda = 3.3\) or better |
| \(= 0.647\) (3 sf) | A1 | SC: Use of Binomial scores B1 for \(0.648\). Use of Normal scores B1 for \(0.67(0)\) to \(0.677\) |
| Total: 3 |
## Question 1:
**Part (i):**
| Answer | Marks | Guidance |
|--------|-------|----------|
| Binomial | B1 | |
| $n = 500$ and $p = \frac{1}{150}$ or $0.00667$ | B1 | Or $B\left(500, \frac{1}{150}\right)$ for B1B1 |
| | **Total: 2** | |
**Part (ii):**
| Answer | Marks | Guidance |
|--------|-------|----------|
| Poisson | B1 | |
| $n$ large and mean $= \frac{10}{3}$ or $3.3$ or better, which is $< 5$ | B1 | Accept $n > 50$ |
| | **Total: 2** | |
**Part (iii):**
| Answer | Marks | Guidance |
|--------|-------|----------|
| $1 - e^{-\frac{10}{3}} \times \left(1 + \frac{10}{3} + \frac{\left(\frac{10}{3}\right)^2}{2}\right)$ | M1 | $1 - P(X = 0, 1, 2)$ |
| $= 1 - 0.353$ | A1 | Correct expression with $\lambda = 3.3$ or better |
| $= 0.647$ (3 sf) | A1 | SC: Use of Binomial scores B1 for $0.648$. Use of Normal scores B1 for $0.67(0)$ to $0.677$ |
| | **Total: 3** | |
---1 On average, 1 in 150 components made by a certain machine are faulty. The random variable $X$ denotes the number of faulty components in a random sample of 500 components.\\
(i) Describe fully the distribution of $X$.\\
(ii) State a suitable approximating distribution for $X$, giving a justification for your choice.\\
(iii) Use your approximating distribution to find the probability that the sample will include at least 3 faulty components.\\
\hfill \mbox{\textit{CAIE S2 2019 Q1 [7]}}