| Exam Board | CAIE |
|---|---|
| Module | Further Paper 4 (Further Paper 4) |
| Year | 2021 |
| Session | November |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Probability Generating Functions |
| Type | Selection without replacement scenarios |
| Difficulty | Challenging +1.2 This is a structured Further Maths probability generating functions question requiring hypergeometric distributions and PGF manipulation. Parts (a) and (b) involve routine calculation of PGFs from first principles using combinatorics, part (c) requires multiplying independent PGFs and expanding, and part (d) is standard differentiation. While the topic is advanced (Further Maths), the question follows a predictable template with clear scaffolding and no novel insights required. |
| Spec | 5.02a Discrete probability distributions: general |
| Answer | Marks | Guidance |
|---|---|---|
| \(P(3,6,9) = \frac{3}{9} \times \frac{2}{8} \times \frac{1}{7} = \frac{6}{504}\) | B1 | At least 2 probabilities correct |
| \(P(\text{Two of } 3,6,9) = \frac{3}{9} \times \frac{2}{8} \times \frac{6}{7} \times 3 = \frac{108}{504}\) | ||
| \(P(\text{one of } 3,6,9) = \frac{3}{9} \times \frac{6}{8} \times \frac{5}{7} \times 3 = \frac{270}{504}\) | ||
| \(P(\text{none of } 3,6,9) = \frac{6}{9} \times \frac{5}{8} \times \frac{4}{7} = \frac{120}{504}\) | ||
| \(G_X(t) = \frac{20}{84} + \frac{45}{84}t + \frac{18}{84}t^2 + \frac{1}{84}t^3\) | M1 A1 | Attempt with at least 3 probabilities in a polynomial, CAO |
| Answer | Marks | Guidance |
|---|---|---|
| \(P(\text{both even}) = \frac{12}{72}\) | M1 | |
| \(P(\text{one even}) = \frac{40}{72}\) | ||
| \(P(\text{no even}) = \frac{20}{72}\) | ||
| \(G_Y(t) = \frac{5}{18} + \frac{10}{18}t + \frac{3}{18}t^2\) | A1 | CAO |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(\frac{1}{1512}\left(\frac{20}{84}+\frac{45}{84}t+\frac{18}{84}t^2+\frac{1}{84}t^3\right)\left(\frac{5}{18}+\frac{10}{18}t+\frac{3}{18}t^2\right)\) | M1 | Method and attempt to multiply |
| Multiplication to obtain single quintic polynomial | M1 | Multiplication to obtain single quintic polynomial |
| \(\frac{1}{1512}(100+425t+600t^2+320t^3+64t^4+3t^5)\) | A1 | CAO |
| Total: 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(G'(1)=\frac{1}{1512}(425+1200+960+256+15)\) | M1 | |
| \(\frac{17}{9}\) | A1 | CWO |
| Total: 2 |
## Question 5(a):
$P(3,6,9) = \frac{3}{9} \times \frac{2}{8} \times \frac{1}{7} = \frac{6}{504}$ | B1 | At least 2 probabilities correct |
$P(\text{Two of } 3,6,9) = \frac{3}{9} \times \frac{2}{8} \times \frac{6}{7} \times 3 = \frac{108}{504}$ | | |
$P(\text{one of } 3,6,9) = \frac{3}{9} \times \frac{6}{8} \times \frac{5}{7} \times 3 = \frac{270}{504}$ | | |
$P(\text{none of } 3,6,9) = \frac{6}{9} \times \frac{5}{8} \times \frac{4}{7} = \frac{120}{504}$ | | |
$G_X(t) = \frac{20}{84} + \frac{45}{84}t + \frac{18}{84}t^2 + \frac{1}{84}t^3$ | M1 A1 | Attempt with at least 3 probabilities in a polynomial, CAO |
## Question 5(b):
$P(\text{both even}) = \frac{12}{72}$ | M1 | |
$P(\text{one even}) = \frac{40}{72}$ | | |
$P(\text{no even}) = \frac{20}{72}$ | | |
$G_Y(t) = \frac{5}{18} + \frac{10}{18}t + \frac{3}{18}t^2$ | A1 | CAO |
## Question 5(c):
| Answer | Mark | Guidance |
|--------|------|----------|
| $\frac{1}{1512}\left(\frac{20}{84}+\frac{45}{84}t+\frac{18}{84}t^2+\frac{1}{84}t^3\right)\left(\frac{5}{18}+\frac{10}{18}t+\frac{3}{18}t^2\right)$ | M1 | Method and attempt to multiply |
| Multiplication to obtain single quintic polynomial | M1 | Multiplication to obtain single quintic polynomial |
| $\frac{1}{1512}(100+425t+600t^2+320t^3+64t^4+3t^5)$ | A1 | CAO |
| **Total: 3** | | |
## Question 5(d):
| Answer | Mark | Guidance |
|--------|------|----------|
| $G'(1)=\frac{1}{1512}(425+1200+960+256+15)$ | M1 | |
| $\frac{17}{9}$ | A1 | CWO |
| **Total: 2** | | |5 Nine balls labelled $1,2,3,4,5,6,7,8,9$ are placed in a bag. Kai selects three balls at random from the bag, without replacement. The random variable $X$ is the number of balls selected by Kai that are labelled with a multiple of 3 .
\begin{enumerate}[label=(\alph*)]
\item Find the probability generating function $\mathrm { G } _ { \mathrm { X } } ( \mathrm { t } )$ of $X$.\\
The balls are replaced in the bag.\\
Jacob now selects two balls at random from the bag, without replacement. The random variable $Y$ is the number of balls selected by Jacob that are labelled with an even number.
\item Find the probability generating function $\mathrm { G } _ { Y } ( \mathrm { t } )$ of $Y$.\\
The random variable $Z$ is the sum of the number of balls that are labelled with a multiple of 3 selected by Kai and the number of balls that are labelled with an even number selected by Jacob.
\item Find the probability generating function of $Z$, expressing your answer as a polynomial.
\item Use the probability generating function of $Z$ to find $\mathrm { E } ( Z )$.
\end{enumerate}
\hfill \mbox{\textit{CAIE Further Paper 4 2021 Q5 [10]}}