3 Morecrest football team always scores at least one goal but never scores more than four goals in each game. The number of goals, \(R\), scored in each game by the team can be modelled by the following probability distribution.
| \(\boldsymbol { r }\) | 1 | 2 | 3 | 4 |
| \(\mathbf { P } ( \boldsymbol { R } = \boldsymbol { r } )\) | \(\frac { 7 } { 16 }\) | \(\frac { 5 } { 16 }\) | \(\frac { 3 } { 16 }\) | \(\frac { 1 } { 16 }\) |
- Calculate exact values for the mean and variance of \(R\).
- Next season the team will play 32 games. They expect to win \(90 \%\) of the games in which they score at least three goals, half of the games in which they score exactly two goals and \(20 \%\) of the games in which they score exactly one goal.
Find, for next season:
- the number of games in which they expect to score at least three goals;
- the number of games that they expect to win.