Two standard six sided dice are to be rolled. If the sum is an even number greater than 7, then what is the probability that both dice are even?

Question

This question is from Mathcounts 2013- 2014, Warm Up 18, Question 253.

1. What formula can you use to find the number of unique dice pairings? *If we have a 3 and a 4, then a 4 and a 3, the two would could as one dice pairing.

I'm thinking that the answer to #1 is 6*2/2. Is this correct?

2. How would you solve the problem as it is written in the title?

Answer 1

The possible outcomes are $$(6,2),\quad (5,3),\quad (4,4),\quad (3,5),\quad (2,6),\quad (6,4),\quad (5,5),\quad (4,6),\quad (6,6)$$ and the successes are $$(6,2),\quad (4,4),\quad (2,6),\quad (6,4),\quad (4,6),\quad (6,6)\ .$$

So the probability is $\frac69=\frac23$.

Note. The somewhat confusing comment in the first question about $4,3$ being the same as $3,4$ is not relevant. You can say that for example $(6,2)$ counts the same as $(2,6)$, but then $6,2$ and $4,4$ will not be equally probable, so you can't just rely on counting outcomes and dividing.