| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2016 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Discrete Probability Distributions |
| Type | Simple algebraic expression for P(X=x) |
| Difficulty | Easy -1.2 This is a straightforward probability distribution question requiring only basic recall: setting up a table, using ΣP(X=x)=1 to find k, then applying standard formulae for mean and variance. All steps are routine with no problem-solving or novel insight required, making it easier than average. |
| Spec | 2.04a Discrete probability distributions2.04b Binomial distribution: as model B(n,p) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(x\): 1, 2, 3, 4 with \(P(x)\): \(k\), \(2k\), \(3k\), \(4k\) | B1 | Probability Distribution Table, either \(k\) or correct numerical values |
| \(10k = 1\) | M1 | Summing probs involving \(k\) to 1, 3 or 4 terms |
| \(k = \dfrac{1}{10}\) | A1 [3] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(E(X) = \dfrac{1}{10} + \dfrac{4}{10} + \dfrac{9}{10} + \dfrac{16}{10} = 3\) | B1 | Correct mean |
| \(\text{Var}(X) = \dfrac{1}{10} + \dfrac{8}{10} + \dfrac{27}{10} + \dfrac{64}{10} - 3^2 = 1\) | M1 A1 [3] | Correct method seen for var, their \(k\) and \(\mu\) |
## Question 3:
### Part (i)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $x$: 1, 2, 3, 4 with $P(x)$: $k$, $2k$, $3k$, $4k$ | B1 | Probability Distribution Table, either $k$ or correct numerical values |
| $10k = 1$ | M1 | Summing probs involving $k$ to 1, 3 or 4 terms |
| $k = \dfrac{1}{10}$ | A1 [3] | |
### Part (ii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $E(X) = \dfrac{1}{10} + \dfrac{4}{10} + \dfrac{9}{10} + \dfrac{16}{10} = 3$ | B1 | Correct mean |
| $\text{Var}(X) = \dfrac{1}{10} + \dfrac{8}{10} + \dfrac{27}{10} + \dfrac{64}{10} - 3^2 = 1$ | M1 A1 [3] | Correct method seen for var, their $k$ and $\mu$ |
---3 A particular type of bird lays 1,2,3 or 4 eggs in a nest each year. The probability of $x$ eggs is equal to $k x$, where $k$ is a constant.\\
(i) Draw up a probability distribution table, in terms of $k$, for the number of eggs laid in a year and find the value of $k$.\\
(ii) Find the mean and variance of the number of eggs laid in a year by this type of bird.
\hfill \mbox{\textit{CAIE S1 2016 Q3 [6]}}