CAIE A-Level Further Mathematics 4.5.1 Probability Generating Functions

Practise constructing a probability generating function from a discrete model, combining independent sums and extracting coefficients, probabilities or moments.

[Maximum number: 5]

Keira has two unbiased coins. She tosses both coins. The number of heads obtained by Keira is denoted by X.

(a)

Find the probability generating function $\mathrm{G}_{X}(t)$ of X.
Hassan has three coins, two of which are biased so that the probability of obtaining a head when the coin is tossed is $\frac{1}{3}$ . The corresponding probability for the third coin is $\frac{1}{4}$ . The number of heads obtained by Hassan when he tosses these three coins is denoted by Y.

[ 1 ]

$\mathrm{G}_{X}(t)=\frac{1}{4}+\frac{1}{2} t+\frac{1}{4} t^{2}$

B1

Accept (0.5+0.5t) ${ }^{2}$

1

(b)

Find the probability generating function $\mathrm{G}_{Y}(t)$ of Y.
The random variable Z is the total number of heads obtained by Keira and Hassan.

[ 3 ]

$\mathrm{P}(0 \mathrm{H})=\frac{12}{36} \quad \mathrm{P}(1 \mathrm{H})=\frac{16}{36} \quad \mathrm{P}(2 \mathrm{H})=\frac{7}{36} \quad \mathrm{P}(3 \mathrm{H})=\frac{1}{36}$

M1 A1

Attempt at probs, at least 2 correct
All correct

$\mathrm{G}_{Y}(t)=\frac{12}{36}+\frac{16}{36} t+\frac{7}{36} t^{2}+\frac{1}{36} t^{3}$

B1 FT

FT their probabilities, must be cubic with 4 non-zero terms

3

(c)

Use the probability generating function of Z to find the most probable value of Z.

[ 1 ]

2

B1 FT

FT power of term with largest coefficient in their $\mathrm{G}_{Z}(t)$

1

All question bank results loaded

4.5.1—Probability generating functions question 1

Question (a)

Question (b)

Question (c)