| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2016 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Binomial Distribution |
| Type | General probability threshold |
| Difficulty | Standard +0.3 This is a straightforward binomial distribution application requiring (i) a direct calculation of P(X ≥ 14) = P(X=14) + P(X=15) with p=0.66, and (ii) solving 0.87^n < 0.04 using logarithms. Both parts are standard textbook exercises with clear setup and routine calculation, slightly above average only due to the two-part structure and need for logarithmic manipulation in part (ii). |
| Spec | 2.04b Binomial distribution: as model B(n,p)2.04c Calculate binomial probabilities |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(p = 0.66\), \(X \sim B(15,\ 0.66)\) | M1 | Bin term \({}^{15}C_x p^x (1-p)^{15-x}\) seen any \(p\) |
| \(P(\text{at least } 14) = P(14,15) = {}^{15}C_{14}(0.66)^{14}(0.34) + (0.66)^{15}\) | M1 | Unsimplified correct expression for \(P(14, 15)\) |
| \(= 0.0171\) | A1 [3] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \((0.87)^n < 0.04\) | M1 | Eqn involving 0.87, power of \(n\), 0.04 only |
| M1 | Solving by logs or trial and error (can be implied). Must be exponential equation | |
| \(n = 24\) | A1 [3] |
## Question 4:
### Part (i)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $p = 0.66$, $X \sim B(15,\ 0.66)$ | M1 | Bin term ${}^{15}C_x p^x (1-p)^{15-x}$ seen any $p$ |
| $P(\text{at least } 14) = P(14,15) = {}^{15}C_{14}(0.66)^{14}(0.34) + (0.66)^{15}$ | M1 | Unsimplified correct expression for $P(14, 15)$ |
| $= 0.0171$ | A1 [3] | |
### Part (ii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $(0.87)^n < 0.04$ | M1 | Eqn involving 0.87, power of $n$, 0.04 only |
| | M1 | Solving by logs or trial and error (can be implied). Must be exponential equation |
| $n = 24$ | A1 [3] | |
---4 When people visit a certain large shop, on average $34 \%$ of them do not buy anything, $53 \%$ spend less than $\$ 50$ and $13 \%$ spend at least $\$ 50$.\\
(i) 15 people visiting the shop are chosen at random. Calculate the probability that at least 14 of them buy something.\\
(ii) $n$ people visiting the shop are chosen at random. The probability that none of them spends at least $\$ 50$ is less than 0.04 . Find the smallest possible value of $n$.
\hfill \mbox{\textit{CAIE S1 2016 Q4 [6]}}