7. The score \(S\) when a spinner is spun has the following probability distribution.
| \(s\) | 0 | 1 | 2 | 4 | 5 |
| \(\mathrm { P } ( S = s )\) | 0.2 | 0.2 | 0.1 | 0.3 | 0.2 |
- Find \(\mathrm { E } ( S )\).
- Show that \(\mathrm { E } \left( S ^ { 2 } \right) = 10.4\)
- Hence find \(\operatorname { Var } ( S )\).
- Find
- \(\mathrm { E } ( 5 S - 3 )\),
- \(\operatorname { Var } ( 5 S - 3 )\).
- Find \(\mathrm { P } ( 5 S - 3 > S + 3 )\)
The spinner is spun twice.
The score from the first spin is \(S _ { 1 }\) and the score from the second spin is \(S _ { 2 }\)
The random variables \(S _ { 1 }\) and \(S _ { 2 }\) are independent and the random variable \(X = S _ { 1 } \times S _ { 2 }\) - Show that \(\mathrm { P } \left( \left\{ S _ { 1 } = 1 \right\} \cap X < 5 \right) = 0.16\)
- Find \(\mathrm { P } ( X < 5 )\).