Probability: chances of winning

[acrimon86]

Homework Statement

Consider a game where a player rolls 2 dice. The player looses if the sum of the dice of the first roll is 2, 3 or 12, and wins if the sum of the dice is 7 or 11. If the outcome of the first roll is 4, 5, 6, 8, 9 or 10, the player continues to roll the dice until the initial outcome is rolled again (player wins) or the 7 appears (the player looses).
Find the probability of winning the game.

Homework Equations

The probability of event E occurring on a sample space $$\Omega$$ is: P(E) = $$\frac{|E|}{|\Omega|}$$

The Attempt at a Solution

The size of the sample space is |$$\Omega$$| = $$6 * 6 = 36$$, since we have two six-sided dice, so a total of 36 possible outcomes on a given throw.

I gather that if the player rolls a sum of 2, 3, or 12 on their first try, they automatically lose the game. Therefore, I let event $$E_{L}$$ denote the event that either of those sums is thrown. Therefore, $$E_{L}$$ is the set of all outcomes that result in one of those sums.

$$E_{L}$$ = { (1,1), (1,1), (2,1), (1,2), (6,6), (6,6) }
|$$E_{L}$$| = 6

Likewise, I let event $$E_{W}$$ denote that a player throws a 7 or 11 on their first try, meaning they automatically win.

$$E_{W}$$ = { (1,6), (6,1), (2,5), (5,2), (3,4), (4,3) }
|$$E_{W}$$| = 6

I also worked out the size of events $$E_{4}$$, $$E_{5}$$, $$E_{6}$$, $$E_{7}$$, $$E_{8}$$, $$E_{9}$$, $$E_{10}$$, where each subscript represents the number of outcomes that could lead to that sum being thrown. I got:

|$$E_{4}$$| = 4
|$$E_{5}$$| = 4
|$$E_{6}$$| = 6
|$$E_{7}$$| = 6
|$$E_{8}$$| = 6
|$$E_{9}$$| = 4
|$$E_{10}$$| = 4

Ok, having that done, the chances of automatically winning the game by throwing a 7 or 11 initially are:
P($$E_{W}$$) = $$\frac{6}{36}$$

This is where I am stuck, however. How do I handle the probabilities of winning by throwing a 4, 5, 6, 8, 9, 10 and then getting that same outcome again without getting a 7?

 

[lippyka]

I know that each of those sums have a certain probability of being thrown initially, and then the same probability of being thrown again before getting a 7. But I'm confused as to how I could express this in terms of an equation/formula. Any help would be greatly appreciated!