| Exam Board | CAIE |
|---|---|
| Module | Further Paper 4 (Further Paper 4) |
| Year | 2023 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Probability Generating Functions |
| Type | Multiple independent coins/dice |
| Difficulty | Standard +0.3 This is a straightforward application of PGF theory for independent random variables. Part (a) requires multiplying three simple binomial PGFs, part (b) multiplies this with another binomial PGF, and part (c) uses the standard derivative formula G'(1). All steps are mechanical with no conceptual challenges beyond knowing the basic PGF formulas. |
| Spec | 5.01a Permutations and combinations: evaluate probabilities5.02a Discrete probability distributions: general |
| Answer | Marks | Guidance |
|---|---|---|
| \(P(3H) = \frac{1}{60}\), \(P(2H) = \frac{9}{60}\), \(P(1H) = \frac{26}{60}\), \(P(0H) = \frac{24}{60}\) | B1 | Two probabilities correct, seen anywhere |
| \(G_X(t) = \frac{24}{60} + \frac{26}{60}t + \frac{9}{60}t^2 + \frac{1}{60}t^3\) | M1 | Cubic polynomial with 4 probabilities as coefficients with the correct powers of \(t\) from their working. Equivalent forms are acceptable |
| A1 | Correct, AEF |
| Answer | Marks | Guidance |
|---|---|---|
| \(G_Y(t) = \frac{1}{4} + \frac{1}{2}t + \frac{1}{4}t^2\) | B1 | |
| \(G_Z(t) = \left(\frac{24}{60} + \frac{26}{60}t + \frac{9}{60}t^2 + \frac{1}{60}t^3\right)\left(\frac{1}{4} + \frac{1}{2}t + \frac{1}{4}t^2\right)\) | M1 | Attempt to multiply their two PGFs |
| \(\frac{1}{240}\left(24 + 74t + 85t^2 + 45t^3 + 11t^4 + t^5\right)\) | M1 | Obtain quintic expression and collect terms |
| A1 | Correct, AEF |
| Answer | Marks | Guidance |
|---|---|---|
| Differentiate: \(G'_Z(t) = \frac{1}{240}\left(74 + 170t + 135t^2 + 44t^3 + 5t^4\right)\) | M1 | Differentiate their \(G\) |
| \(G'_Z(1) = \frac{428}{240} = \frac{107}{60} = 1.78\) | A1 | Any correct form |
## Question 5(a):
| $P(3H) = \frac{1}{60}$, $P(2H) = \frac{9}{60}$, $P(1H) = \frac{26}{60}$, $P(0H) = \frac{24}{60}$ | B1 | Two probabilities correct, seen anywhere |
|---|---|---|
| $G_X(t) = \frac{24}{60} + \frac{26}{60}t + \frac{9}{60}t^2 + \frac{1}{60}t^3$ | M1 | Cubic polynomial with 4 probabilities as coefficients with the correct powers of $t$ from their working. Equivalent forms are acceptable |
| | A1 | Correct, AEF |
---
## Question 5(b):
| $G_Y(t) = \frac{1}{4} + \frac{1}{2}t + \frac{1}{4}t^2$ | B1 | |
|---|---|---|
| $G_Z(t) = \left(\frac{24}{60} + \frac{26}{60}t + \frac{9}{60}t^2 + \frac{1}{60}t^3\right)\left(\frac{1}{4} + \frac{1}{2}t + \frac{1}{4}t^2\right)$ | M1 | Attempt to multiply their two PGFs |
| $\frac{1}{240}\left(24 + 74t + 85t^2 + 45t^3 + 11t^4 + t^5\right)$ | M1 | Obtain quintic expression and collect terms |
| | A1 | Correct, AEF |
---
## Question 5(c):
| Differentiate: $G'_Z(t) = \frac{1}{240}\left(74 + 170t + 135t^2 + 44t^3 + 5t^4\right)$ | M1 | Differentiate their $G$ |
|---|---|---|
| $G'_Z(1) = \frac{428}{240} = \frac{107}{60} = 1.78$ | A1 | Any correct form |
---5 Harry has three coins.
\begin{itemize}
\item One coin is biased so that, when it is thrown, the probability of obtaining a head is $\frac { 1 } { 3 }$.
\item The second coin is biased so that, when it is thrown, the probability of obtaining a head is $\frac { 1 } { 4 }$.
\item The third coin is biased so that, when it is thrown, the probability of obtaining a head is $\frac { 1 } { 5 }$.
\end{itemize}
The random variable $X$ is the number of heads that Harry obtains when he throws all three coins together.
\begin{enumerate}[label=(\alph*)]
\item Find the probability generating function of $X$.\\
Isaac has two fair coins. The random variable $Y$ is the number of heads that Isaac obtains when he throws both of his coins together. The random variable $Z$ is the total number of heads obtained when Harry throws his three coins and Isaac throws his two coins.
\item Find the probability generating function of $Z$, expressing your answer as a polynomial in $t$.
\item Use the probability generating function of $Z$ to find $E ( Z )$.
\end{enumerate}
\hfill \mbox{\textit{CAIE Further Paper 4 2023 Q5 [9]}}