CAIE A-Level Further Mathematics 4.5 Probability Generating Functions Question Bank

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

B1

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

1

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

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

$\mathrm{G}_{Z}(t)=\left(\frac{1}{4}+\frac{1}{2} t+\frac{1}{4} t^{2}\right)\left(\frac{12}{36}+\frac{16}{36} t+\frac{7}{36} t^{2}+\frac{1}{36} t^{3}\right)$

M1

Attempt to multiply their two PGF

$=\frac{1}{144}\left(12+40 t+51 t^{2}+31 t^{3}+9 t^{4}+t^{5}\right)$

M1 A1

Obtain quintic expression and collect terms

3

$\mathrm{G}_{Z}^{\prime}(t)=\frac{1}{144}\left(40+102 t+93 t^{2}+36 t^{3}+5 t^{4}\right)$

M1

Differentiate

$\mathrm{E}(Z)=\mathrm{G}_{Z}^{\prime}(1)=\frac{23}{12}=(=1.92)$

A1

2

2

B1 FT

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

1