1. a) 5 girls and 3 boys are arranged at random in a straight line. Find the probability that none of the boys is standing next to another boy.
   (3 marks)
   b) A cricket team consisting of six batsmen, four bowlers, and one wicket-keeper is to be selected from a group of 18 cricketers comprising nine batsmen, seven bowlers, and two wicket-keepers.
   How many different teams can be selected?
   (3 marks)
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2. \(\quad \mathrm { P } ( E ) = 0.25 , \mathrm { P } ( F ) = 0.4\) and \(\mathrm { P } ( E \cap F ) = 0.12\)
   a Find \(P \left( E ^ { \prime } \mid F ^ { \prime } \right)\)
   b Explain, showing your working, whether or not \(E\) and \(F\) are statistically independent. Give reasons for your answer.

The event \(G\) has \(\mathrm { P } ( G ) = 0.15\)
The events \(E\) and \(G\) are mutually exclusive and the events \(F\) and \(G\) are independent.
c Draw a Venn diagram to illustrate the events \(E , F\) and \(G\), giving the probabilities for each region.
d Find \(\mathrm { P } \left( [ F \cup G ] ^ { \prime } \right)\)
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