Suppose a game is designed as follows.Each player that enters the gameis asked to draw two ball from an identical bag that consists of 15 balls: 8 red balls, 4 blue balls and 3 green balls. Assume that the bag is not transparent so that the players cannot see the balls in the bag when drawing.

• The player wins $8 if he draws two blue balls.
• The player wins $5 if he draws two green balls.
• The player loses $3 if he draws one blue ball and one green ball.
• The player loses $10 if he draws at least one red ball.

Here we define the profit as the amount of dollars the player wins or loses, in which the profit is larger than $0 if the player wins dollars and the profit is less than $0 if the player loses dollars.

(a) Find the moment generating function of the profit of a randomly chosen player.[4 marks]

(b) Based on the moment generating function that you found in part (a), find the variance of the profit of a randomly chosen player. (You will not be given any mark if you and the expectation without relating the expectation to the moment generating function).[4 marks]

(c) Suppose 10 players are randomly chosen to play the game, and assume that the outcomes of the 5 players are independent of each other. Find the moment generating function of the total profit (sum of the profit) of the 5 randomly chosen players.[4 marks]