| Exam Board | CAIE |
|---|---|
| Module | Further Paper 4 (Further Paper 4) |
| Year | 2022 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Probability Generating Functions |
| Type | Solve for parameters using PGF coefficients |
| Difficulty | Challenging +1.2 This is a standard PGF question requiring setup of the generating function from a probability distribution, solving simultaneous equations from coefficients, and applying standard PGF properties (composition for sums, variance from derivatives). While it involves Further Maths content, the techniques are routine applications of PGF theory with straightforward algebra—no novel insight required. |
| Spec | 5.01a Permutations and combinations: evaluate probabilities |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(G_X(t) = \frac{6}{12} + \frac{5}{12}t + \frac{1}{12}t^2\); \(P(\text{0 heads}) = (1-a)(1-b) = \frac{6}{12}\); \(P(\text{1 head}) = a(1-b)+(1-a)b = \frac{5}{12}\); \(P(\text{2 heads}) = ab = \frac{1}{12}\); giving \(\left(a+b = \frac{7}{12},\ ab = \frac{1}{12}\right)\) | M1 | Obtain two correct equations and attempt to solve |
| \(a = \frac{1}{3}\) \(\left(b = \frac{1}{4}\right)\) | A1 | Correct value for \(a\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(G_Y(t) = \left(\frac{6}{12} + \frac{5}{12}t + \frac{1}{12}t^2\right)^2\) | M1 | Square their \(G_X(t)\) |
| \(\frac{1}{144}\left(36 + 60t + 37t^2 + 10t^3 + t^4\right)\) | M1 | Obtain quartic polynomial from 3-term \(G_X(t)\) |
| A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(G_Y'(t) = \frac{1}{144}\left(60 + 74t + 30t^2 + 4t^3\right)\); \(G_Y''(t) = \frac{1}{144}\left(74 + 60t + 12t^2\right)\) | M1 | Differentiate their \(G_Y(t)\) twice |
| \(\text{Var}(Y) = G_Y''(1) + G_Y'(1) - (G_Y'(1))^2\); \(\frac{1}{144}(146) + \frac{7}{6} - \frac{49}{36} = \frac{59}{72} = 0.819\) | M1 | Use correct formula and attempt at \(\text{Var}(Y)\) |
| \(0.819\) | A1 |
## Question 3(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $G_X(t) = \frac{6}{12} + \frac{5}{12}t + \frac{1}{12}t^2$; $P(\text{0 heads}) = (1-a)(1-b) = \frac{6}{12}$; $P(\text{1 head}) = a(1-b)+(1-a)b = \frac{5}{12}$; $P(\text{2 heads}) = ab = \frac{1}{12}$; giving $\left(a+b = \frac{7}{12},\ ab = \frac{1}{12}\right)$ | M1 | Obtain two correct equations and attempt to solve |
| $a = \frac{1}{3}$ $\left(b = \frac{1}{4}\right)$ | A1 | Correct value for $a$ |
---
## Question 3(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $G_Y(t) = \left(\frac{6}{12} + \frac{5}{12}t + \frac{1}{12}t^2\right)^2$ | M1 | Square their $G_X(t)$ |
| $\frac{1}{144}\left(36 + 60t + 37t^2 + 10t^3 + t^4\right)$ | M1 | Obtain quartic polynomial from 3-term $G_X(t)$ |
| | A1 | |
---
## Question 3(c):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $G_Y'(t) = \frac{1}{144}\left(60 + 74t + 30t^2 + 4t^3\right)$; $G_Y''(t) = \frac{1}{144}\left(74 + 60t + 12t^2\right)$ | M1 | Differentiate their $G_Y(t)$ twice |
| $\text{Var}(Y) = G_Y''(1) + G_Y'(1) - (G_Y'(1))^2$; $\frac{1}{144}(146) + \frac{7}{6} - \frac{49}{36} = \frac{59}{72} = 0.819$ | M1 | Use correct formula and attempt at $\text{Var}(Y)$ |
| $0.819$ | A1 | |
---3 George throws two coins, $A$ and $B$, at the same time. Coin $A$ is biased so that the probability of obtaining a head is $a$. Coin $B$ is biased so that the probability of obtaining a head is $b$, where $\mathrm { b } < \mathrm { a }$. The probability generating function of $X$, the number of heads obtained by George, is $\mathrm { G } _ { \mathrm { X } } ( \mathrm { t } )$. The coefficients of $t$ and $t ^ { 2 }$ in $\mathrm { G } _ { \mathrm { X } } ( \mathrm { t } )$ are $\frac { 5 } { 12 }$ and $\frac { 1 } { 12 }$ respectively.
\begin{enumerate}[label=(\alph*)]
\item Find the value of $a$.\\
The random variable $Y$ is the sum of two independent observations of $X$.
\item Find the probability generating function of $Y$, giving your answer as a polynomial in $t$.
\item Find $\operatorname { Var } ( Y )$.
\end{enumerate}
\hfill \mbox{\textit{CAIE Further Paper 4 2022 Q3 [8]}}