This one's actually a little tricky. What you seem to be calculating is the probability that you'll get a sum of 3 on the first throw, and a sum of 7 on the second throw (or vice versa). What you actually want is the probability that you attain a sum of 3 before a sum of 7.

To begin with, determine the respective probabilities:

(sum of 3 on any roll of the dice)

(sum of 7 on any roll of the dice)

(You should be able to obtain these easily by counting the outcomes).

Now, we want the probability of attaining sum of 3 before sum of 7. This is equivalent to the probability that, when you obtain thefirstsum of 3, your previous rolls have not had sums of 7 (or 3, obviously). In this sense we have what resembles a geometric random variable:

(first sum of 3 obtained on the roll, without obtaining any sums of 7 before the roll) .

This is of course for only some value of . Summing over all possible values of , we get:

(first sum of 3 obtained without obtaining sums of 7 before)