Frank is a cold-hearted economist who has been given a chance to enter a 2-die lottery with an amazing prize. The lottery works as follows. Before two fair dice are rolled, Frank can choose an integer number n between 2 and 12, that is 2 n 12. If the sum of the rolled dice matches Frank's selection (e.g., the roll is (5, 5) and his selection was 10), then Frank wins the prize. Otherwise he loses. Before he makes a decision, Frank wants to contemplate this problem in the light of his expected utility model. For that, he needs to construct the outcome space and assign probabilities to the different numbers he can select.

1. Write down the outcome space, i.e., the space of all possible valid outcomes in this lottery. For example, a valid outcome is the draw (1, 4), which signifies that the roll of the first die ended in 1 and the roll of the second ended in 4.
2. What is the probability that Frank wins if he chooses n = 5?
3. What is the probability that Frank loses if he chooses n = 10?