| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2019 |
| Session | November |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Binomial Distribution |
| Type | General probability threshold |
| Difficulty | Moderate -0.3 Part (i) is a straightforward binomial probability calculation requiring P(X < 8) = 1 - P(X ≥ 8) with n=10, p=0.42. Part (ii) requires setting up 1-(0.58)^n > 0.995 and solving for n using logarithms. Both parts are standard S1 binomial questions with routine techniques, though part (ii) requires slightly more algebraic manipulation than typical, keeping it just below average difficulty overall. |
| Spec | 2.04b Binomial distribution: as model B(n,p)2.04c Calculate binomial probabilities |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(1 - ({}^{10}C_2\, 0.42^8 0.58^2 + {}^{10}C_9\, 0.42^9 0.58^1 + 0.42^{10})\) | M1 | Binomial term of form \({}^{10}C_a p^a(1-p)^b\), \(0 < p < 1\) any \(p\), \(0 \leq a,b \leq 10\) |
| A1 | Correct unsimplified expression | |
| \(0.983\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(1 - P(0) > 0.995 \Rightarrow 0.58^n < 0.005\) | M1 | Equation or inequality involving \(0.58^n\) or \(0.42^n\) and 0.995 or 0.005 |
| \(n > \dfrac{\log 0.005}{\log 0.58}\), \(n > 9.727\) | M1 | Attempt to solve using logs or Trial and Error. May be implied by their answer (rounded or truncated) |
| \(n = 10\) | A1 | CAO |
## Question 2(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $1 - ({}^{10}C_2\, 0.42^8 0.58^2 + {}^{10}C_9\, 0.42^9 0.58^1 + 0.42^{10})$ | M1 | Binomial term of form ${}^{10}C_a p^a(1-p)^b$, $0 < p < 1$ any $p$, $0 \leq a,b \leq 10$ |
| | A1 | Correct unsimplified expression |
| $0.983$ | A1 | |
**Total: 3 marks**
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## Question 2(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $1 - P(0) > 0.995 \Rightarrow 0.58^n < 0.005$ | M1 | Equation or inequality involving $0.58^n$ or $0.42^n$ and 0.995 or 0.005 |
| $n > \dfrac{\log 0.005}{\log 0.58}$, $n > 9.727$ | M1 | Attempt to solve using logs or Trial and Error. May be implied by their answer (rounded or truncated) |
| $n = 10$ | A1 | CAO |
**Total: 3 marks**
---2 Annan has designed a new logo for a sportswear company. A survey of a large number of customers found that $42 \%$ of customers rated the logo as good.\\
(i) A random sample of 10 customers is chosen. Find the probability that fewer than 8 of them rate the logo as good.\\
(ii) On another occasion, a random sample of $n$ customers of the company is chosen. Find the smallest value of $n$ for which the probability that at least one person rates the logo as good is greater than 0.995 .\\
\hfill \mbox{\textit{CAIE S1 2019 Q2 [6]}}