Odds of flopping 2 pair or better

[Ragnarok]

I'm quite curious to know what are the odds to get a better hand than a pair on the flop, which doesn't seem to develop often enough. Could someone help me or at least point me to a website/post explaining this.

Also, I'd very much appreciate if someone could answer a stranger question: How often should a hand statistically stay unimproved, just a high card.

I'd very much appreciate answers to the questions, thanks a lot in advance.

[Bugsbunny]

The first question is pretty wide open - ir really depends on what your cards are. If they're conncted then they'll improve more often than if they're not. A maximal connecter will improve more often than a 1 or 2 gap etc.

However looking just at pairs with 2 unsuited, unconnected, cards it would go something like this:

you either need trips (with the board pairing and hitting you) or you need the board to hit you twice.

50 cards in the deck (plus your 2) so you have 19600 possible flops:

The board will pair and hit one of your cards:
3 ways to pair + 47 possible kickers (this includes boats) = 3 * 47 = 141 possible combinations. You have 2 cards so that gives 282 combinations total.

quads are 2 additional combinations so 284 total.

next we have the case where you get exactly 2 pair. You have 23 of each card available. 3 * 3 = 9 combinations with 44 possible kickers. 9 * 44 = 396

396 + 284 = 680

680/19600= 0.0346938776, or about 3.5% of the time

Connectors and suited cards add possibilities for straights and flushes so they'll be higher.

As for the second question - again ifnoring straights and flushes and just looking at pairs:
Given the same situation - 2 unpaired, unconnected, unsuited, cards:
50 cards taken 5 at a time = 2,118,760

Ignoring the possibilities of the board pairing (or better) and just looking at how often you won't at least pair (also ignoring straights and flushes):
There are 44 cards in the deck that can hit that won't touch your cards.
44 taken 5 at a time = 1,086,008
1086008/2118760=0.512567728, so just over 51% of the time the board will miss you completely (again ignoring straights or flushes).