15.3 Exercise 10

A game is played as follows:

1. A wheel is spun, giving one of the integers 0 through 9 on an equally likely basis.
2. A single die is thrown the number of times indicated by the result of the spin of the wheel. The number of points made is the total of the numbers turned up on the sequence of throws of the die.
3. A player pays sixteen dollars to play; a dollar is returned for each point made.

Let Y represent the number of points made and $X = Y - 16$ be the net gain (possibly negative) of the player. Determine the maximum value of

X, $E \left[X\right]$, $Var \left[X\right]$, $P \left(X > 0\right)$, $P \left(X > = 10\right)$, $P \left(X > = 16\right)$.