A spinner is designed so that the score \(S\) is given by the following probability distribution.
\(s\)
0
1
2
4
5
\(\mathrm { P } ( S = s )\)
\(p\)
0.25
0.25
0.20
0.20
Find the value of \(p\).
Find \(\mathrm { E } ( S )\).
Show that \(\mathrm { E } \left( S ^ { 2 } \right) = 9.45\)
Find \(\operatorname { Var } ( S )\).
Tom and Jess play a game with this spinner. The spinner is spun repeatedly and \(S\) counters are awarded on the outcome of each spin. If \(S\) is even then Tom receives the counters and if \(S\) is odd then Jess receives them. The first player to collect 10 or more counters is the winner.
Find the probability that Jess wins after 2 spins.
Find the probability that Tom wins after exactly 3 spins.
Find the probability that Jess wins after exactly 3 spins.