I made this calculation mainly as a consideration for when you can/should bet an overpair on a paired board.

Assuming completely random hands, here's what I came up with if you yourself flopped trips:

If there are 6 OTHER players in the hand, there's a probability of 53% that someone else also has trips. Less than that, and in all probability no one else has trips.

But if you don't have trips, then with 3 OTHER players in, the probability is 46% that someone has them. With 4, it goes up to 59%, and with only 2, the probability is 32%.

In terms of betting a naked overpair on a paired board, of course, there are lots of other factors that come into play--like the quality both of the trips and of the side-cards as well as whether someone might play back at you light and such. But purely in terms of the likelihood of being ahead, low cards are certainly good for your hand, particularly 2s and 3s, which are just less common in playable starting hands for anyone who has at least some hand selection going on.

I also think a big side-card makes things more dangerous, since a full house becomes another possibility.

But I do think betting AA or KK is pretty reasonable into a field of 3 other players on a board of T22 or such. If it's Q22, it starts to get a little iffy, and I'd probably go down to betting it maybe only with position against that many players (against 2, I'd still do it). Also, on a big card board, you can often half-pot and reduce your investment by repping QQ on the Q22 board.

Anyhow, my conclusion from these probabilities is that you're reasonably safe on paired boards in general in a field of only 2 other players. If there are 3 others, you have to look at the specific board and ideally have a bit of a read. Against 4 or more other players, I think as a rule you simply need to have hit the flop harder than just having an overpair.