| Exam Board | CAIE |
|---|---|
| Module | Further Paper 4 (Further Paper 4) |
| Year | 2024 |
| Session | November |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Probability Generating Functions |
| Type | Multiple independent coins/dice |
| Difficulty | Standard +0.8 This is a standard Further Maths probability generating functions question requiring knowledge of PGF for independent random variables and their sums. Part (a) requires multiplying three simple PGFs, part (b) extends this by adding another independent variable, and part (c) uses differentiation. While methodical, it's routine application of PGF theory with straightforward arithmetic—typical for Further Maths but above standard A-level. |
| Spec | 5.02a Discrete probability distributions: general |
| ΝΙΟθW SΙΗΙ ΝΙ ΞιΙΥΜ ιΟΝ Ο0 | \includegraphics[max width=\textwidth, alt={}]{e2a45d19-7d48-4aa5-93f9-6ef90f99d7c4-10_446_37_674_2013} | \includegraphics[max width=\textwidth, alt={}]{e2a45d19-7d48-4aa5-93f9-6ef90f99d7c4-10_444_37_1245_2013} | \includegraphics[max width=\textwidth, alt={}]{e2a45d19-7d48-4aa5-93f9-6ef90f99d7c4-10_441_33_1816_2013} | \includegraphics[max width=\textwidth, alt={}]{e2a45d19-7d48-4aa5-93f9-6ef90f99d7c4-10_443_33_2387_2013} |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(P(0 \text{ heads}) = \frac{8}{30}\), \(P(1 \text{ head}) = \frac{14}{30}\), \(P(2 \text{ heads}) = \frac{7}{30}\), \(P(3 \text{ heads}) = \frac{1}{30}\) | B1 | All correct |
| \(G_X(t) = \frac{8}{30} + \frac{14}{30}t + \frac{7}{30}t^2 + \frac{1}{30}t^3\) | M1 | Cubic polynomial |
| A1 FT | FT their probabilities that sum to one |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(G_Y(t) = \frac{25}{36} + \frac{10}{36}t + \frac{1}{36}t^2\) | B1 | |
| \(G_Z(t) = \frac{1}{30 \times 36}\left(\frac{8}{30} + \frac{14}{30}t + \frac{7}{30}t^2 + \frac{1}{30}t^3\right)(25 + 10t + t^2)\) | M1 | With attempt to multiply |
| \(\frac{1}{1080}(200 + 430t + 323t^2 + 109t^3 + 17t^4 + t^5)\) | M1 A1 | Obtain quintic polynomial |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(G_Z'(1) = \frac{1}{1080}(430 + 646 + 327 + 68 + 5)\) | M1 | Differentiate and substitute \(t = 1\) |
| \(\frac{1476}{1080} = \frac{41}{30} = 1.37\) | A1 |
## Question 5(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $P(0 \text{ heads}) = \frac{8}{30}$, $P(1 \text{ head}) = \frac{14}{30}$, $P(2 \text{ heads}) = \frac{7}{30}$, $P(3 \text{ heads}) = \frac{1}{30}$ | B1 | All correct |
| $G_X(t) = \frac{8}{30} + \frac{14}{30}t + \frac{7}{30}t^2 + \frac{1}{30}t^3$ | M1 | Cubic polynomial |
| | A1 FT | FT their probabilities that sum to one |
## Question 5(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $G_Y(t) = \frac{25}{36} + \frac{10}{36}t + \frac{1}{36}t^2$ | B1 | |
| $G_Z(t) = \frac{1}{30 \times 36}\left(\frac{8}{30} + \frac{14}{30}t + \frac{7}{30}t^2 + \frac{1}{30}t^3\right)(25 + 10t + t^2)$ | M1 | With attempt to multiply |
| $\frac{1}{1080}(200 + 430t + 323t^2 + 109t^3 + 17t^4 + t^5)$ | M1 A1 | Obtain quintic polynomial |
## Question 5(c):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $G_Z'(1) = \frac{1}{1080}(430 + 646 + 327 + 68 + 5)$ | M1 | Differentiate and substitute $t = 1$ |
| $\frac{1476}{1080} = \frac{41}{30} = 1.37$ | A1 | |
---5 Nikita has three coins. One coin is fair, one coin is biased so that the probability of obtaining a head is $\frac { 1 } { 3 }$ and the third coin is biased so that the probability of obtaining a head is $\frac { 1 } { 5 }$. The random variable $X$ is the number of heads that Nikita obtains when he throws all three coins at the same time.
\begin{enumerate}[label=(\alph*)]
\item Find the probability generating function of $X$.\\
Rajesh has two fair six-sided dice with faces labelled 1, 2, 3, 4, 5, 6. The random variable $Y$ is the number of 4 s that Rajesh obtains when he throws the two dice.
The random variable $Z$ is the sum of the number of heads obtained by Nikita and the number of 4 s obtained by Rajesh.
\item Find the probability generating function of $Z$, expressing your answer as a polynomial.\\
\begin{center}
\begin{tabular}{|l|l|l|l|l|}
\hline
ΝΙΟθW SΙΗΙ ΝΙ ΞιΙΥΜ ιΟΝ Ο0 & \includegraphics[max width=\textwidth, alt={}]{e2a45d19-7d48-4aa5-93f9-6ef90f99d7c4-10_446_37_674_2013}
& \includegraphics[max width=\textwidth, alt={}]{e2a45d19-7d48-4aa5-93f9-6ef90f99d7c4-10_444_37_1245_2013}
& \includegraphics[max width=\textwidth, alt={}]{e2a45d19-7d48-4aa5-93f9-6ef90f99d7c4-10_441_33_1816_2013}
& \includegraphics[max width=\textwidth, alt={}]{e2a45d19-7d48-4aa5-93f9-6ef90f99d7c4-10_443_33_2387_2013}
\\
\hline
\end{tabular}
\end{center}
\includegraphics[max width=\textwidth, alt={}, center]{e2a45d19-7d48-4aa5-93f9-6ef90f99d7c4-11_2726_35_97_20}
\item Use your answer to part (b) to find $\mathrm { E } ( Z )$.
\end{enumerate}
\hfill \mbox{\textit{CAIE Further Paper 4 2024 Q5 [9]}}