| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2004 |
| Session | November |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Combinations & Selection |
| Type | Probability distributions from selection |
| Difficulty | Standard +0.3 This is a straightforward combinatorics problem requiring systematic listing of outcomes and basic probability calculations. While part (iv) requires computing expectation and variance from a distribution, all steps are routine applications of standard formulas with no novel problem-solving insight needed. The small sample space (10 outcomes) makes enumeration manageable, placing this slightly above average difficulty due to the multiple parts and careful bookkeeping required. |
| Spec | 5.01a Permutations and combinations: evaluate probabilities5.02a Discrete probability distributions: general5.02b Expectation and variance: discrete random variables |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Options 123, 124, 125, 134, 135, 145, 234, 235, 245, 345 | M1 | For listing options, at least 4 different ones |
| \(P(\text{odd}) = 0.4\) | B1 2 | For correct answer, legit obtained |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(P(\text{largest is } 4) = 0.3\) | B1 1 | For correct answer |
| OR \(\frac{1 \times {}_3C_2}{{}_5C_3}\) | SR if 9 options in (i) give B1 for 3/9 or 2/9 depending on their missing option |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(l \quad\quad 3 \quad\quad 4 \quad\quad 5\) | M1 | For 3, 4, 5 in table or 1, 2 as well; no need for any probs but need to see an (uncompleted) second line |
| \(P(L=l) \quad 0.1 \quad 0.3 \quad 0.6\) | M1 | For evaluating another probability based on their list |
| A1 3 | For correct answer |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(E(L) = \sum lp = 3 \times 0.1 + 4 \times 0.4 + 5 \times 0.6 = 4.5\) | B1ft | For correct answer, ft if their \(\sum p = 1\) |
| \(\text{Var}(L) = 3^2 \times 0.1 + 4^2 \times 0.3 + 5^2 \times 0.6 - (\text{their } 4.5^2)\) | M1 | For evaluating their \(\sum l^2 p - (\text{their } 4.5^2)\) (must see \(- \text{their } 4.5^2\)) each \(p < 1\), in first numerical instance, ie can forget the sq rt subsequently |
| \(= 0.45\) | A1 3 | For correct answer |
# Question 6:
## Part (i)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Options 123, 124, 125, 134, 135, 145, 234, 235, 245, 345 | M1 | For listing options, at least 4 different ones |
| $P(\text{odd}) = 0.4$ | B1 **2** | For correct answer, legit obtained |
## Part (ii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $P(\text{largest is } 4) = 0.3$ | B1 **1** | For correct answer |
| OR $\frac{1 \times {}_3C_2}{{}_5C_3}$ | | **SR** if 9 options in **(i)** give B1 for 3/9 or 2/9 depending on their missing option |
## Part (iii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $l \quad\quad 3 \quad\quad 4 \quad\quad 5$ | M1 | For 3, 4, 5 in table or 1, 2 as well; no need for any probs but need to see an (uncompleted) second line |
| $P(L=l) \quad 0.1 \quad 0.3 \quad 0.6$ | M1 | For evaluating another probability based on their list |
| | A1 **3** | For correct answer |
## Part (iv)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $E(L) = \sum lp = 3 \times 0.1 + 4 \times 0.4 + 5 \times 0.6 = 4.5$ | B1ft | For correct answer, ft if their $\sum p = 1$ |
| $\text{Var}(L) = 3^2 \times 0.1 + 4^2 \times 0.3 + 5^2 \times 0.6 - (\text{their } 4.5^2)$ | M1 | For evaluating their $\sum l^2 p - (\text{their } 4.5^2)$ (must see $- \text{their } 4.5^2$) each $p < 1$, in first numerical instance, ie can forget the sq rt subsequently |
| $= 0.45$ | A1 **3** | For correct answer |
---6 A box contains five balls numbered $1,2,3,4,5$. Three balls are drawn randomly at the same time from the box.\\
(i) By listing all possible outcomes (123, 124, etc.), find the probability that the sum of the three numbers drawn is an odd number.
The random variable $L$ denotes the largest of the three numbers drawn.\\
(ii) Find the probability that $L$ is 4 .\\
(iii) Draw up a table to show the probability distribution of $L$.\\
(iv) Calculate the expectation and variance of $L$.
\hfill \mbox{\textit{CAIE S1 2004 Q6 [9]}}