| Exam Board | CAIE |
|---|---|
| Module | Further Paper 4 (Further Paper 4) |
| Year | 2020 |
| Session | Specimen |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Probability Generating Functions |
| Type | Find PGF from probability distribution |
| Difficulty | Standard +0.3 This is a straightforward application of PGF theory with sampling with replacement. Part (a) requires finding a binomial distribution PGF (standard formula), part (b) is identical, part (c) uses the independence property G_Z(t) = G_X(t)·G_Y(t), and part (d) applies standard PGF differentiation formulas for mean and variance. All techniques are direct applications of bookwork with no novel problem-solving required. Slightly easier than average due to the repetitive structure and mechanical nature. |
| Spec | 5.02a Discrete probability distributions: general5.02b Expectation and variance: discrete random variables |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Probabilities \(\frac{3}{8}, \frac{3}{8}, \frac{1}{8}, \frac{1}{8}\) for 0, 1, 2, 3 reds | B1 | |
| \(G_X(t) = \frac{1}{8} + \frac{3}{8}t + \frac{3}{8}t^2 + \frac{1}{8}t^3\) | B1FT | Follow through their probabilities so long as \(\Sigma p = 1\) |
| 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Correct value \(\frac{9}{20}\) or 0.45 for \(P(Y=1)\) or \(P(Y=2)\) | B1 | |
| Probabilities \(\frac{1}{20}, \frac{9}{20}, \frac{9}{20}, \frac{1}{20}\) for 0, 1, 2, 3 reds | B1 | |
| \(G_Y(t) = \frac{1}{20} + \frac{9}{20}t + \frac{9}{20}t^2 + \frac{1}{20}t^3\) | B1FT | Follow through their probabilities so long as \(\Sigma p = 1\) |
| 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(G_Z(t) = G_X(t) \times G_Y(t)\) | B1 | Stated or implied |
| Complete expansion of the product of their cubics | M1 | Or equivalent, e.g. with explicit fractional coefficients |
| \(G_Z(t) = \frac{1}{160}(1 + 12t + 39t^2 + 56t^3 + 39t^4 + 12t^5 + t^6)\) | A1 | |
| 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Attempt to differentiate \(G_Z(t)\) and evaluate \(G'_Z(1)\) | M1 | |
| \(E(Z) = 3\) | A1 | Correctly obtained from \(G'_Z(1)\) |
| Attempt to find second derivative \(G''_Z(t)\) | M1 | |
| Use of \(G'_Z(1) + G''_Z(1) - (G'_Z(1))^2\) | M1 | |
| \(\text{Var}(Z) = 1.2\) | A1 | Correct value correctly obtained |
| 5 |
# Question 6:
## Part 6(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Probabilities $\frac{3}{8}, \frac{3}{8}, \frac{1}{8}, \frac{1}{8}$ for 0, 1, 2, 3 reds | **B1** | |
| $G_X(t) = \frac{1}{8} + \frac{3}{8}t + \frac{3}{8}t^2 + \frac{1}{8}t^3$ | **B1FT** | Follow through their probabilities so long as $\Sigma p = 1$ |
| | **2** | |
## Part 6(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Correct value $\frac{9}{20}$ or 0.45 for $P(Y=1)$ or $P(Y=2)$ | **B1** | |
| Probabilities $\frac{1}{20}, \frac{9}{20}, \frac{9}{20}, \frac{1}{20}$ for 0, 1, 2, 3 reds | **B1** | |
| $G_Y(t) = \frac{1}{20} + \frac{9}{20}t + \frac{9}{20}t^2 + \frac{1}{20}t^3$ | **B1FT** | Follow through their probabilities so long as $\Sigma p = 1$ |
| | **3** | |
## Part 6(c):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $G_Z(t) = G_X(t) \times G_Y(t)$ | **B1** | Stated or implied |
| Complete expansion of the product of their cubics | **M1** | Or equivalent, e.g. with explicit fractional coefficients |
| $G_Z(t) = \frac{1}{160}(1 + 12t + 39t^2 + 56t^3 + 39t^4 + 12t^5 + t^6)$ | **A1** | |
| | **3** | |
## Part 6(d):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Attempt to differentiate $G_Z(t)$ and evaluate $G'_Z(1)$ | **M1** | |
| $E(Z) = 3$ | **A1** | Correctly obtained from $G'_Z(1)$ |
| Attempt to find second derivative $G''_Z(t)$ | **M1** | |
| Use of $G'_Z(1) + G''_Z(1) - (G'_Z(1))^2$ | **M1** | |
| $\text{Var}(Z) = 1.2$ | **A1** | Correct value correctly obtained |
| | **5** | |6 Aisha has a bag containing 3 red balls and 3 white balls. She selects a ball at random, notes its colour and returns it to the bag; the same process is repeated twice more. The number of red balls selected by Aisha is denoted by $X$.
\begin{enumerate}[label=(\alph*)]
\item Find the probability generating function $\mathrm{G}_{X}(t)$ of $X$.
\end{enumerate}
Basant also has a bag containing 3 red balls and 3 white balls. He selects three balls at random, without replacement, from his bag. The number of red balls selected by Basant is denoted by $Y$.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item Find the probability generating function $\mathrm{G}_{Y}(t)$ of $Y$.
\end{enumerate}
The random variable $Z$ is the total number of red balls selected by Aisha and Basant.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{2}
\item Find the probability generating function of $Z$, expressing your answer as a polynomial.
\item Use the probability generating function of $Z$ to find $\mathrm{E}(Z)$ and $\operatorname{Var}(Z)$.
\end{enumerate}
\hfill \mbox{\textit{CAIE Further Paper 4 2020 Q6 [13]}}