SDW2 wrote:
Is there any other approach to this question?
VeritasKarishma IanStewart egmatThe fastest way to solve (and I'm really just saying what Bunuel said, but in different words) is to notice:
• when we roll one die, we're just as likely to get an above-average roll as a below-average one. That's what the earlier posts mean when they say dice outcomes are "symmetric". The same is true when we roll three dice.
• when we roll one die, the average roll is 3.5 (it's the average of the equally spaced list 1, 2, 3, 4, 5, 6, so it is the median of that list). So when we roll three dice, the average roll is (3)(3.5) = 10.5. Since we can't get exactly 10.5, since that's not an integer, and since we're as likely to get an above average sum as a below average one, we must get a sum greater than 10.5 exactly half the time, and a sum less than 10.5 the other half of the time. So we get 11 or greater 1/2 the time.
There's an obvious alternative approach, but it takes a long time: you can just list all of the ways to get a sum of 18 (we must get 6-6-6), and a sum of 17 (we can get 5-6-6, 6-5-6 or 6-6-5), and so on. Using certain counting shortcuts makes that perhaps practical to do in two minutes, but it's long, and it's too tedious to be the only available method in a real GMAT question. There are other ways one could solve that are similar -- e.g. you could divide the problem into cases where you imagine getting a '6' on the first die, then a '5' on the first die, and so on, but they aren't any faster. I don't see an alternative solution to the first one I suggest above that is at all fast.