| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2017 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Poisson distribution |
| Type | Single time period probability |
| Difficulty | Moderate -0.3 This is a straightforward application of the Poisson distribution with clearly stated parameter λ=1.4. Parts (i) and (ii) require direct formula substitution (P(X=2) and 1-P(X=0)), while part (iii) involves solving an inequality using logarithms—all standard S1 techniques with no conceptual challenges or novel problem-solving required. Slightly easier than average due to its routine nature. |
| 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 |
|---|---|---|
| Answer | Marks | Guidance |
| \(p = 0.07\) | B1 | |
| \(P(2) = {}^{20}C_2 (0.07)^2 (0.93)^{18}\) | M1 | Bin term \({}^{20}C_x p^x (1-p)^{20-x}\) their \(p\) |
| \(= 0.252\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(P(\text{at least 1 cracked egg}) = 1 - (0.93)^{20} = 1 - 0.2342\) | M1 | Attempt to find \(P(\text{at least 1 cracked egg})\) with their \(p\) from (i), allow \(1 - P(0,1)\) OE |
| \(= 0.766\) | A1 | Rounding to 0.766 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \((0.7658)^n < 0.01\) | M1 | Equation or inequality containing \((0.766)^n\) or \((0.234)^n\), together with 0.01 or 0.99 |
| \(n = 18\) | A1 |
## Question 5(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $p = 0.07$ | B1 | |
| $P(2) = {}^{20}C_2 (0.07)^2 (0.93)^{18}$ | M1 | Bin term ${}^{20}C_x p^x (1-p)^{20-x}$ their $p$ |
| $= 0.252$ | A1 | |
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## Question 5(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $P(\text{at least 1 cracked egg}) = 1 - (0.93)^{20} = 1 - 0.2342$ | M1 | Attempt to find $P(\text{at least 1 cracked egg})$ with their $p$ from (i), allow $1 - P(0,1)$ OE |
| $= 0.766$ | A1 | Rounding to 0.766 |
---
## Question 5(iii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $(0.7658)^n < 0.01$ | M1 | Equation or inequality containing $(0.766)^n$ or $(0.234)^n$, together with 0.01 or 0.99 |
| $n = 18$ | A1 | |
---5 Eggs are sold in boxes of 20. Cracked eggs occur independently and the mean number of cracked eggs in a box is 1.4 .\\
(i) Calculate the probability that a randomly chosen box contains exactly 2 cracked eggs.\\
(ii) Calculate the probability that a randomly chosen box contains at least 1 cracked egg.\\
(iii) A shop sells $n$ of these boxes of eggs. Find the smallest value of $n$ such that the probability of there being at least 1 cracked egg in each box sold is less than 0.01 .\\
\hfill \mbox{\textit{CAIE S1 2017 Q5 [7]}}