| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2021 |
| Session | March |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Discrete Probability Distributions |
| Type | Simple algebraic expression for P(X=x) |
| Difficulty | Moderate -0.5 This is a straightforward probability distribution question requiring basic algebraic manipulation. Part (a) involves simple substitution into the given formula, and part (b) requires standard variance calculation using E(X²) - [E(X)]². While it involves multiple steps, the techniques are routine and commonly practiced in S1, making it slightly easier than average. |
| Spec | 2.04a Discrete probability distributions5.02b Expectation and variance: discrete random variables |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(x\): 1, 2, 3, 4; prob: \(4k\), \(6k\), \(6k\), \(4k\) | B1 | Table with \(x\) values and one correct probability expressed in terms of \(k\). Condone any additional \(x\) values if probability stated as 0 |
| Remaining 3 probabilities correct | B1 | Remaining 3 probabilities correct expressed in terms of \(k\) – condone if the first correct probability is not in table |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \([4k+6k+6k+4k=1]\ k=\frac{1}{20}\ (=0.05)\) | B1 | Correct value for \(k\) SOI. May be calculated in 4(a). SC B1 if denominator \(20k\) used throughout |
| \(E(X)=1\times\frac{4}{20}+2\times\frac{6}{20}+3\times\frac{6}{20}+4\times\frac{4}{20}=\frac{4+12+18+16}{20}\ (=2.5)\) | M1 | Accept unsimplified expression. Condone \(4k+12k+18k+16k\). May be implied by use in Variance expression. Special ruling: Allow use of denominator \(20k\) |
| \(\text{Var}(X) = 1^2\times\frac{4}{20}+2^2\times\frac{6}{20}+3^2\times\frac{6}{20}+4^2\times\frac{4}{20}-\left(their\ 2\frac{1}{2}\right)^2\) | M1 | Appropriate variance formula with *their* numerical probabilities using *their* \((E(X))^2\), accept unsimplified, with *their* \(k\) substituted. Special ruling: If denominator \(20k\) used throughout, accept appropriate variance formula in terms of \(k\) |
| \(1.05\) | A1 | AG, NFWW |
## Question 4(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $x$: 1, 2, 3, 4; prob: $4k$, $6k$, $6k$, $4k$ | B1 | Table with $x$ values and one correct probability expressed in terms of $k$. Condone any additional $x$ values if probability stated as 0 |
| Remaining 3 probabilities correct | B1 | Remaining 3 probabilities correct expressed in terms of $k$ – condone if the first correct probability is not in table |
## Question 4(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $[4k+6k+6k+4k=1]\ k=\frac{1}{20}\ (=0.05)$ | B1 | Correct value for $k$ SOI. May be calculated in 4(a). SC B1 if denominator $20k$ used throughout |
| $E(X)=1\times\frac{4}{20}+2\times\frac{6}{20}+3\times\frac{6}{20}+4\times\frac{4}{20}=\frac{4+12+18+16}{20}\ (=2.5)$ | M1 | Accept unsimplified expression. Condone $4k+12k+18k+16k$. May be implied by use in Variance expression. **Special ruling:** Allow use of denominator $20k$ |
| $\text{Var}(X) = 1^2\times\frac{4}{20}+2^2\times\frac{6}{20}+3^2\times\frac{6}{20}+4^2\times\frac{4}{20}-\left(their\ 2\frac{1}{2}\right)^2$ | M1 | Appropriate variance formula with *their* numerical probabilities using *their* $(E(X))^2$, accept unsimplified, with *their* $k$ substituted. **Special ruling:** If denominator $20k$ used throughout, accept appropriate variance formula in terms of $k$ |
| $1.05$ | A1 | AG, NFWW |4 The random variable $X$ takes the values $1,2,3,4$ only. The probability that $X$ takes the value $x$ is $k x ( 5 - x )$, where $k$ is a constant.
\begin{enumerate}[label=(\alph*)]
\item Draw up the probability distribution table for $X$, in terms of $k$.
\item Show that $\operatorname { Var } ( X ) = 1.05$.
\end{enumerate}
\hfill \mbox{\textit{CAIE S1 2021 Q4 [6]}}