Use an area model or a tree diagram to represent the sample space for rolling two standard six-sided dice.

1. In the standard casino dice game the roller wins on the first roll if he rolls a sum of $7$ or $11$. What is the probability of winning on the first roll?

   Hint (a):

   There are $6$ ways to get a sum of $7$, and $2$ ways to get $11$.

   Answer (a):

   $P(7,\text{ or },11)=\frac{8}{36}=\frac{2}{9}$

2. The player loses on the first roll if he rolls a sum of $2$, $3$, or $12$. What is the probability of losing on the first roll?

   Answer (b):

   $\frac{1}{9}$

3. If the player rolls any other sum, he continues to roll the dice until that sum comes up again or until he rolls a 7, whatever happens sooner. What is the probability that the game continues after the first roll?

   Answer (c):

   $\frac{2}{3}$