The GameMaster's Poker School
Lesson 25: Is My Ace Dominated?

When I got my first computer back in 1996, I wondered where the huge number-crunching ability all computers were supposed to have could be found. I somehow envisioned myself figuring the value of pi to 7 million decimal places, but had no idea if the computer could do it, or even how to go about making such a calculation, even if my machine was capable of doing so. Well, of course computers can do stuff like that. For me, it's done in the amazing program called Excel by Microsoft and now that I (finally!) know how to use Excel, it crunches all kinds of numbers, far beyond my imagination.

Although this article is about Poker, you're going to see an entire series of articles eventually appear here that will fall under the umbrella of "Excel For the Advantage Player". Most people don't like the mathematical aspects of casino gaming as much as I do, but by showing you how to use Excel, most of the pain associated with the math will hopefully be eliminated. Calculations like standard deviation, variance, combinatorial analysis and other mathematical functions are performed by Excel in literally the blink of an eye, if you know how to program it correctly.

And that's what I'll show you here: how to program Excel to give you the answers you seek. For starters, I thought it would be interesting to see just what the chances are that your hand of Ace-whatever is "dominated" by another A-x hand (the "x" means any other card). By dominated, I mean that you and an opponent share the Ace, but his kicker - the second card in his hand - may be higher than yours. Of course, if you already have an A-x hand, the probability that your opponent also has an A-x hand is reduced because the deck contains only three Aces at that point, instead of the usual four. But, it's still possible for him to have it and if you have any experience with the game at all, you know players will hold on to their Aces with a death grip. I wrote about that in Lesson 22, and this lesson will supplement that by giving you some solid numbers as to the probability your hand is the best when comparing A-x hands.

Knowing the probability that your A-x hand is best is important at anytime, but I think it's crucial to know this once you get to the point in a tournament where you're facing just one other opponent. Needless to say, in a heads up situation, any hand with an Ace in it is capable of winning the match, but there's still the possibility that your opponent has you dominated, so before you go all-in at least know what your odds are.

Before I get into the specific chart, let me show you a few simple calculations that Excel can do when it comes to Poker hands. The vast majority use the combination function, which is COMBIN in Excel programming language.

Common Poker Calculations

To make a calculation, just open a blank Excel worksheet and select the "cell" (like B3 or D2) where you want the answer to appear. When you select the cell, a dark box will appear around it. Then, type the equal sign (=), which tells Excel you're going to give it a formula. Let's say you want to know how many 5-card hands can be made by a 52-card deck of playing cards. After the = sign, you enter COMBIN, then a left side paren, ( , enter the appropriate numbers and close with a rightside paren, ). It must look like this to work: =COMBIN(52,5), although you don't have to use caps on the words - I just do that to show it's a formula. The number on the left is the "population" available for your selection. The number after the comma is the number you want to select from the population. When you hit "Enter", the number 2598960 should appear in the cell you selected. The formula will appear in the formula bar that's at the top of the worksheet, so you'll know how the number in that cell was calculated. If you want to show the number as 2,598,960, just right-click the cell and choose "Format cells". That will take you to a menu of choices, one of which is Number. Choose that and click on "Use 1000 Separator (,)" if you want commas in your numbers. You can also choose the number of decimal places up to 30, depending upon your needs.

As you can see, there are 2,595,960 different 5-card Poker hands in each 52-card deck. If you want to know how many two-card hands are possible, enter the formula, =COMBIN(52,2) and you'll get an answer of 1,326. Of course, this formula takes suits in account, so 2c-Jh is a different hand than 2h-Jc, but most of the time we don't really care; 2-J is 2-J and A-A is A-A, which of course can never be suited. If we want to know just how many distinct two-card hands are possible without regard to suits, the formula is a bit more complicated, but it does give you an insight to some of the basic Poker calculations Excel can perform. As you know, a standard Poker deck has 13 "ranks" of cards - Ace to King (or 2 to Ace if you prefer), each of which are in four "suits" - spades, hearts, clubs and diamonds - and that totals 52. The number of suited two-card combinations for one suit, say spades, is COMBIN(13,2) or 78 - 13 cards of the same suit, from which we pick any two. Each of the 13 ranks of cards have four each, but they're of different suits when they form a pair. Each rank can combine to form pairs in COMBIN(4,2) or 6 ways. (Get it? The 4 is the "population" - like four Queens - and the 2 is the number, a pair, you want to select from that population.) Preflop, Qc-Qs is the same as Qh-Qd so we count only the 13 pairs as being distinct hands: 2-2 to A-A. And, because we really don't care if we get As-2s or Ah-2h; they're both A-2 preflop, 78 combinations are suited and 78 are not (like Ah-2s or As-2h). Thus, there are a total of 78 + 78 + 13 = 169 distinct two-card hands available in a deck of cards.

In every game of Hold'em, you'll begin with one of those 169 possible "hole cards". Whether or not they're suited or a pair might make a difference in how you play them, but most of the time they won't be either. Of the 1,326 possible two-card hands (those 169 starting hands but taking suits into consideration), only COMBIN(4,2)*13 - four cards of one rank made into a pair multiplied by 13 ranks - or 78 will be the magical pocket pair. Thus, the odds of receiving any poket pair is 78 in 1326 or 1 in 17 or 16 to 1 against, which translates into 1/17 or 0.056%. The formula for that last number is =1/17. A note here: In Excel, a "-" is minus, a "+" is plus, a "*" is multiply and a "/" is divide. The odds of receiving a specific pocket pair, such as Aces is COMBIN(4,2) or 6 of 1326, which is 1 in 221 or 220 to 1 against. which translates into 1/221 or 0.045%.

Obviously, the vast majority of the time your hole cards will be of two different ranks, such as A-K. We know there are four Aces and four Kings in a deck, so we can use Excel to tell us how many different hands those eight cards can make with the formula =COMBIN(8,2), which is 28. Of those 28 combinations, some are pairs that we'll want to exclude if we need to know how many hands of A-K are possible. We've seen that before: =COMBIN(4,2) or 6, but because we're dealing with both Aces and Kings, we need to multiply it by 2. So, the number of pairs of Aces or Kings that can be made from a deck of cards is =COMBIN(4,2)*2 or 12, which tells us there are 28 minus 12 or 16 ways a hand of A-K can be made preflop, some of which are suited; most of which are not. Naturally, if you have A-K as your starting cards, the probability that one of your opponents also has it is reduced, because the deck is now only 50 cards and has just 3 Aces and 3 Kings in it. (Hopefully you understand that it doesn't change the odds if you get your cards first or your opponent does; in a random deck, any card unseen is still considered to be available for play.) But in the case where you have A-K, it's a fact that only 3 Aces and 3 Kings remain. Thus, your opponent's probability of also holding a hand made up of some combination of Aces and/or Kings is COMBIN(6,2) or 15. Of those, COMBIN (3,2)*2 or 6 are pairs; three pair of Aces and three pairs of Kings. That leaves only 9 hands of A-K available to your opponents. Needless to say, you'd hate to see them have A-A or K-K, but I'll talk about pairs a little later.

The Dominated Hand Chart

Let me now set the stage for the chart I'll present to answer the probability your hand of A-x is "dominated." As your kicker gets lower, the probability that you're hand is dominated increases. Whether or not your hand is dominated is also affected by the number of players at the table - the more hands that are dealt, the more opportunity there is for someone to catch an A-x better than yours. The hand of A-K is dominated only by A-A, but a hand of A-Q is dominated by A-K and A-A, so A-J is dominated by A-Q, A-K and A-A and so forth. All we need to do now is total up the number of hands involved to develop a percentage probability, which I prefer to use over odds in situations like this. (I know that a 1 in 20 chance is five percent probability, but I get confused when it's presented as 19 to 1 against.) We already know that there is the same opportunity to receive a hand of A-5 as A-K; both are calculated as COMBIN (8,2), with the 8 representing a population of four Aces, plus four other cards, and the 2 representing the number of choices we wish to make out of the population.

That being the case, it's really just an exercise of addition to see how many possible hands of those remaining can beat yours. Now remember that the deck is changed when your two hole cards are removed. Your heads-up opponent is now playing with a 50-card deck that contains one less Ace and one less x, whatever x is, because they're now in your hand and unavailable for play. If your x is a 10, they have less of a chance of catching a 10, but a better chance of catching a King because all four are still in the deck. Get the idea of how this works? Here's something else you need to know. Because you have two cards, the number of possible starting hands for one opponent has changed to COMBIN(50,2) or 1225, which is the number I used for calculating the probabilities in the Heads Up play portion of my chart.

For every combination of A-x where seven cards are available - three Aces and four "others", there are 21 possible combinations: COMBIN(7,2). Of those, 3 are pairs of Aces: COMBIN(3,2) and 6 are pairs of "others": COMBIN(4,2). Thus, there are 21-3-6 = 12 A-x hands available for each rank where you do not hold one of the "other" cards. If you have A-Q in the hole, there are 12 A-K and 12 A-J hands, for example, that can be made from the deck, but if you have A-K, now there are only 3 Aces and 3 Kings remaining and they'll make - as I mentioned earlier - only 9 A-K hands to compete with yours. But because we really don't care about ties here, those can be removed from our probabilities. So, a hand of A-K is dominated only by A-A, which is just 3 hands of 1225 and now you know why Big Slick is such a good non-pair starting hand. (Of course running into K-K is no fun either, but it's not a "dominating" hand in the strictest sense of the word.)

One further note before you look at the numbers. Don't confuse what I'm presenting here with the probability of winning the hand percentages that you see all the time on televised Poker tournaments. I think most of us know that two overcards like A-K is a slight underdog to say, a pair of 9s - about 44 to 56 percent. My numbers show the probability, expressed in percentages, that a better hand than yours is out there, preflop. A hand of A-2 has a 24 percent chance of beating A-K, a 5% chance of a tie and a 71% chance of losing, but those aren't the percentages I'm presenting. My numbers do not predict who will win the hand; that's up to the poker gods. My numbers show only the probability that (A) another A-x hand is out there and (B) that hand dominates yours.

Okay, here's my chart: 

Developing the numbers for Heads Up was easy. I just calculated the number of dominating hands and divided by 1225. In the case of A-K, there are only 3 hands of A-A possible, so the percentage is 3 divided by 1225 (=3/1225) or 0.22%. Thus, only about one time out of 400 times that you hold A-K will you run into A-A when playing Heads Up. For the hand of A-Q, it's dominated by the same A-A (3 hands) and all hands of A-K (12 hands), so the probability it's dominated is 15/1225 or 1.22%, about one in 80. By definition, A-2 is dominated by all of the other A-x hands, but even that humble hand is about a 90% favorite to be best before the flop, assuming your opponent doesn't hold a pocket pair.

However, when we move to multiple opponents, playing a hand like A-2 has its risks even though not every player will have a hand of A-x. (The maximum is three of course because there are only three Aces left in the deck.) To calculate these percentages is a really long and involved process, but luckily someone already did it for 10 opponents. If you wish to see the very complicated formula, plus pick up some other great statistics on poker, go here: http://www.math.sfu.ca/~alspach/computations.html/. Article # 28 is what you want. A tip of the ol' GameMaster's hat to Brian Alspach for making this available and granting permission to use it.

Using this chart

This is another one of those charts that you should keep handy as you play Poker online. I don't think you need to memorize it, but you should have a good "working knowledge" of it - at least know that calling with A-8 on the button means there's a 40% probability that your hand is dominated by a higher A-x hand, so the pot odds ought to be about 2 to 1 before you call. Also, if you get into Heads Up play (which you might do after watching the National Heads Up Poker Championship that will air on NBC in April, 2006), this chart can save or make you a lot of $$$. If you want to modify the chart by adding pairs to it - a thought that occurred to me as my hand of A-10 got beat by 10-10 in a tournament today, just add back the number of available pairs for each rank (usually 12 if you're not already holding one of them) and recalculate the percentages using, ta da! Excel. Speaking of pairs, The Wizard of Odds has a chart that shows the probability that your pocket pair is dominated by a higher pair, depending upon the number of players in the hand - from two to ten. It's a chart I have sitting by me as I play Hold 'em. You can find it here: http://wizardofodds.com/askthewizard/146/