View Full Version : Odds calculations in "Encyclopedia of Draw Poker", by Anno

04-25-2005, 12:54 PM
I originally posted this in the "Other Poker" forum because it deals with draw, but someone suggested posting it here as well, so here it is...

I picked up a copy of this book a little while ago. It has lots of odds calculations about various starting hands and draw results, including an analysis of the effects of keeping a kicker when drawing to a pair. When drawing three to a pair he gives the probability of improving to a given better hand as approximately:

two-pair = 1/6
trips = 1/9

These seem right to me, but he then looks at keeping a kicker and drawing two and says the chances in that case change to:

two-pair = 1/12
trips = 1/26

Neither of these seems right to me, so I though I'd give my reasoning and let some of the math guys (of which I used to be one in the distant past) tell me if I'm right or wrong.

Let's write the two possible situations (draw-3, draw-2) like this:


In the two-pair case, it seems intuitive to me that your chances are no worse if you keep a kicker than if you draw three. My reasoning is that in the draw-three case, what does the first card you draw have to be to keep alive your chances of drawing exactly to 2-pair? All it has to do is not be an A, the chances of which are 45/47 (2 A's remaining, 47 cards left after your 2 A's and the three discards). Now you have AACxx, and from this point you're in exactly the same situation as if you'd kept the kicker, having to draw one more C in the last two cards. So it seems to me that the chances of getting to exactly two-pair by drawing two are very slightly greater than by drawing three, by a ratio of 47/45.

In the case of improving to exactly trips you need to draw exactly one A in each case. When you have a few events whose probabilities are very small you can get a decent approximation of the chance of hitting one by adding the probabilities up. Since this case fits that condition (and ignoring the very small probabilities of drawing to a full house of quads) the chance of getting trips when drawing two should be pretty close to 2/3 the chance of getting it by drawing three, not less than half as good as the book states that it is.

Can anyone confirm (or refute) my reasoning here?

04-25-2005, 04:10 PM
Suppose you start with just one pair.

If you just keep a pair, there are 16215 possibilities. There are

11559 ways to make 1 pair,
2592 ways to make 2 pair,
1854 ways to make 3 of a kind,
165 ways to make a full house, and
45 ways to make 4 of a kind.

If you keep a kicker, there are 1081 possibilities. There are

801 ways to make one pair,
186 ways to make two pair,
84 ways to make 3 of a kind,
9 ways to make a full house, and
1 way to make 4 of a kind.

Rather than doing the calculations myself, I used WinPoker, software for analyzing video poker. These look right to me, though. The book appears to be quite wrong.