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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:

AAxxx
AABxx

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?

pzhon
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.