2 In a game at a charity fair, a player rolls 3 unbiased six-sided dice. The random variable \(X\) represents the difference between the highest and lowest scores.
- Show that \(\mathrm { P } ( X = 0 ) = \frac { 1 } { 36 }\).
The table shows the probability distribution of \(X\).
| \(r\) | 0 | 1 | 2 | 3 | 4 | 5 |
| \(\mathrm { P } ( \mathrm { X } = \mathrm { r } )\) | \(\frac { 1 } { 36 }\) | \(\frac { 5 } { 36 }\) | \(\frac { 2 } { 9 }\) | \(\frac { 1 } { 4 }\) | \(\frac { 2 } { 9 }\) | \(\frac { 5 } { 36 }\) |
- Draw a graph to illustrate the distribution.
- Describe the shape of the distribution.
- In this question you must show detailed reasoning.
Find each of the following.
- \(\mathrm { E } ( X )\)
- \(\operatorname { Var } ( X )\)
As a result of playing the game, the player receives \(30 X\) pence from the organiser of the game. - Find the variance of the amount that the player receives.
- The player pays \(k\) pence to play the game.
Given that the average profit made by the organiser is 12.5 pence per game, determine the value of \(k\).