Three of a Kind in Draw

[JustPlayingSmart]

I was doing some basic probability tables for draw poker, and I noticed that when you are dealt three-of-a-kind before the draw, and draw two cards, your chances of getting four-of-a-kind are double that (4.26%) if you were drawing with three-of-a-kind and a kicker (2.13%). In addition, your chances of getting either a full house or quads are 10.36% when drawing two, while only 8.51% if drawing one. Given that the kicker is irrelevant in this situation (it wouldn't come into play because only one player can have three-of-a-kind for a given rank), would there ever be a reason to draw one card instead of two here?

[JustPlayingSmart]

I also performed this analysis on holding a pair and kicker (drawing two) to just a pair (drawing three). The results were that when drawing three, you make 2 pair or better 28.71% of the time, while you only make 2 pair or better 25.90% of the time if you draw two. It doesn't seem like this difference is enough to compensate for the times when you have an equal pair with someone and discarded top kicker. I am not sure of the math behind that, however.

[Nottom]

The only reason I can see to keep extra cards in this situation is for deception purposes. It certainly doesn't give you a better chance to improve.

[lunchmeat]

It is sometimes correct to draw less than the optimal number of cards, especially against 1 or 2 tight opponents who can be easily bluffed. You basically have to weigh the chances of your bluff knocking out a better hand compared to the advantage you lose by drawing fewer cards.

[Al Mirpuri]

False carding, that is, misrepresenting your hand by not drawing the maximum number of cards possible for your hand, is a big part of good Draw strategy. You would draw 1 card when you had one or two callers. Anyone making two pair would pay you off when you rebet your trips after the draw. A maniac making Aces Up would give you excessive action. Against more than two callers you would probably only keep the kicker with high trips as any caller after the draw would probably beat low trips. So with low trips you are looking to improve with the draw.

[Al Mirpuri]

Keeping the kicker with a pair is a definite mistake. Improving becomes less likely and you gain little by way of deception. You would disguise trips by drawing one card to them not by drawing two cards to a pair (if you had had trips then you would have raised before the draw).

As for having a pair equal with someone else's and throwing away the high kicker being a mistake, it is only rarely so. You would need to have a pair of Aces and be convinced your opponent has a pair of Aces and so keeping your King kicker would be correct. This is the only pair versus pair scenario where keeping a kicker would be correct.

It is impossible to know and mathematically rare to have a pair exactly the same as someone else's and for you both to be leading the table before the draw. except in the scenario outlined above.

At draw, keeping a kicker with a pair, is a sure sign that a player is a weak player.