In the limiting case the number of sixes will be larger than 1/6 half the time, smaller than 1/6 half the time, and exactly 1/6 only an infinitesimal amount of

In the limiting case the number of sixes will be larger than 1/6 half the time, smaller than 1/6 half the time, and exactly 1/6 only an infinitesimal amount of the time. Roughly speaking, with a smaller number of rolls you are more likely to get at least 1/6 sixes because the "exactly 1/6" case becomes more significant. However, this is a bit too rough because just because the expected number of sixes is 1/6 it doesn't follow (and isn't true) that you are equally likely to get more than 1/6 and less than 1/6 sixes. You are actually more likely to get less than 1/6, just not by enough to offset the chance of getting exactly 1/6. You can get intuition by just looking at a particular small case, like the six throws case, or to make the numbers even simpler you could look at three rolls of a three-sided die.