| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2013 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Binomial Distribution |
| Type | Identify distribution and parameters |
| Difficulty | Moderate -0.3 This is a straightforward binomial distribution question requiring identification of parameters (counting multiples of 5 from 7-21 gives p=3/15=1/5), a routine probability calculation using tables or calculator, and solving an inequality involving (4/5)^n < 0.01. All steps are standard S1 techniques with no novel insight required, making it slightly easier than average. |
| Spec | 2.04b Binomial distribution: as model B(n,p)2.04c Calculate binomial probabilities |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(X \sim \text{Bin}(12, 0.2)\) | B1 | Bin or B |
| B1 | 12 | |
| B1 [3] | 0.2 or \(\frac{1}{5}\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(P(X = 3, 4, 5) = 0.2^3 0.8^9 \,{}_{12}C_3 + 0.2^4 0.8^8 \,{}_{12}C_4 + 0.2^5 0.8^7 \,{}_{12}C_5\) | M1 | Bin expression with any \(p\) |
| \(= 0.23622 + 0.13287 + 0.05315\) | A1ft | Correct unsimplified expression, their \(p\) |
| \(= 0.422\) | A1 [3] | Correct answer |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(P(X = 0) < 0.01\) | M1 | Statement involving \(P(X=0)\) and 0.01 can be implied |
| \(0.8^n < 0.01\) | M1 | Equation involving '0.8', 0.01 or 0.99 |
| \(n = 21\) | A1 [3] | Correct answer |
## Question 5:
### Part (i):
| Answer | Mark | Guidance |
|--------|------|----------|
| $X \sim \text{Bin}(12, 0.2)$ | B1 | Bin or B |
| | B1 | 12 |
| | B1 **[3]** | 0.2 or $\frac{1}{5}$ |
### Part (ii):
| Answer | Mark | Guidance |
|--------|------|----------|
| $P(X = 3, 4, 5) = 0.2^3 0.8^9 \,{}_{12}C_3 + 0.2^4 0.8^8 \,{}_{12}C_4 + 0.2^5 0.8^7 \,{}_{12}C_5$ | M1 | Bin expression with any $p$ |
| $= 0.23622 + 0.13287 + 0.05315$ | A1ft | Correct unsimplified expression, their $p$ |
| $= 0.422$ | A1 **[3]** | Correct answer |
### Part (iii):
| Answer | Mark | Guidance |
|--------|------|----------|
| $P(X = 0) < 0.01$ | M1 | Statement involving $P(X=0)$ and 0.01 can be implied |
| $0.8^n < 0.01$ | M1 | Equation involving '0.8', 0.01 or 0.99 |
| $n = 21$ | A1 **[3]** | Correct answer |
---5 Fiona uses her calculator to produce 12 random integers between 7 and 21 inclusive. The random variable $X$ is the number of these 12 integers which are multiples of 5 .\\
(i) State the distribution of $X$ and give its parameters.\\
(ii) Calculate the probability that $X$ is between 3 and 5 inclusive.
Fiona now produces $n$ random integers between 7 and 21 inclusive.\\
(iii) Find the least possible value of $n$ if the probability that none of these integers is a multiple of 5 is less than 0.01.
\hfill \mbox{\textit{CAIE S1 2013 Q5 [9]}}