1. (a) (i) Name a suitable distribution for modelling the volume of liquid in bottles of wine sold as containing 75 cl .
   (ii) Explain why the mean in such a model would probably be greater than 75 cl .
   (b) (i) Name a suitable distribution for modelling the score on a single throw of a fair four-sided die with the numbers \(1,2,3\) and 4 on its faces.
   (ii) Use your suggested model to find the mean and variance of the score on a single throw of the die.
   (6 marks)
2. The events \(A\) and \(B\) are independent and such that

$$\mathrm { P } ( A ) = 2 \mathrm { P } ( B ) \text { and } \mathrm { P } ( A \cap B ) = \frac { 1 } { 8 }$$ (a) Show that \(\mathrm { P } ( B ) = \frac { 1 } { 4 }\).
(b) Find \(\mathrm { P } ( A \cup B )\).
(c) Find \(\mathrm { P } \left( A \mid B ^ { \prime } \right)\).