1 A fair five-sided spinner has sectors labelled 1, 2, 3, 4, 5. In a game at a stall at a charity event, the spinner is spun twice. The random variable \(X\) represents the lower of the two scores. The probability distribution of \(X\) is given by the formula
\(\mathrm { P } ( \mathrm { X } = \mathrm { r } ) = \mathrm { k } ( 11 - 2 \mathrm { r } )\) for \(r = 1,2,3,4,5\),
where \(k\) is a constant.
- Complete the copy of this table in the Printed Answer Booklet.
| \(r\) | 1 | 2 | 3 | 4 | 5 |
| \(\mathrm { P } ( X = r )\) | | \(7 k\) | | \(3 k\) | |
- Determine the value of \(k\).
- Find each of the following.
- \(\mathrm { E } ( X )\)
- \(\operatorname { Var } ( X )\)
- The stall-holder charges a player \(C\) pence to play the game, and then pays the player \(50 X\) pence, where \(X\) is the player's score.
Given that the average profit that the stall-holder makes on one game is 25 pence, find the value of \(C\).