[AaronBrown]

The 2/3's statistic means the chance of pairing (or better) one of two distinct pocket cards. If you start with a pair, your chance of missing are greater since there are only two cards that help you instead of six. A pair turns into a set or better only 12% of the time. With connectors or suited cards, "hitting" usually means getting a four flush or open-end straight draw or better.

It's pretty close to true that the chance of two different players hitting the flop is independent. As you say, it depends on the overlap between the hands, and it also depends on the type of flop (three of a kind on the flop, for example, can only be hit by one player at most, and that can happen in only one way). But in the long run, these effects will average out.

Therefore, the chance of no one hitting the flop is about (2/3)^N, where N is the number of players. The list below shows for N=1 to 10, the probability of no one hitting the flop, exactly one player hitting the flop and more than one player hitting the flop.

1 68% 32% 0%
2 46% 44% 11%
3 31% 44% 25%
4 21% 40% 39%
5 14% 34% 52%
6 10% 27% 63%
7 6% 22% 72%
8 4% 17% 79%
9 3% 13% 84%
10 2% 10% 88%