Lowball Poker Probabilities

What, exactly, constitutes a "bad hand?" Consider the chances of being dealt a good hand and the chances of obtaining hand improvement in a Lowball draw. The math is much the same as for Five-Card Draw. The probability of not being dealt a hand with a pair of Jacks or higher ("openers" in Five-Card Draw) is 79%. The probability of not being dealt a pair of any kind or a higher hand is 50.1%. For Lowball, additional questions are relevant: What is the probability of being dealt a hand with four unpaired and unsuited cards of eight or under? Three cards? Then, if I draw one card, what are the chances that it will help me? If I draw two cards? Three?

The probabilities of being dealt a seven high hand (irrespective of straights or flushes) are less than 0.002%. For eight high, the number is 0.005%. For nine, ten and Jack, the probabilities are 1.3%, 2.7%, and 4.9% respectively. This means that most of the time a player will not have a hand that can sail to showdown without some improvement. This is also true, of course, in high-card draw poker as well. Your skill in betting and then choosing the right draw is what makes you a winner.

Now if an opponent is standing pat after the deal, you know that the chances that he or she has a seven-high or eight-high hand are quite remote - less than one in a hundred. Many players new to Lowball will stand pat with somewhat high hands if they are rough - probable losers, but drawing won't help much, if at all. In a full table game, for example, a rough Jack hand is fairly common. It may win once in a long while, but most pots will go to seven-high or maybe even eight-high hands in Deuce-to-Seven Lowball. In Ace-to-Five, anything over seven- or eight-high is a probable loser.

The calculation for getting a good draw hand is as follows. The desired hand is, let's say, four cards of eight or less, unpaired (for Deuce-to-Seven) or sevens or under (for Ace-to-Five). First, we disregard the problem of straights and flushes, as four in a row or in the same suit is just fine for now. In Ace-to-Five Lowball it never matters, and in Deuce-to-Seven Lowball, the breaking up of a straight or a flush comes in considering outs on the draw, if the hand is subject to becoming a straight or a flush. The number of possible deals is the usual 2.6 million.

Without the restrictions of suit and sequence (but honoring the rule against pairs), the chances of being dealt five cards that match this criterion are 0.591%. There are 7 possible ranks. One is pre-specified as the "high" (in this case, the eight) and can be one of four suits. The other four cards in the hand (anything in six ranks from deuce to seven) can be arrayed in combin(6,4) combinations in one of four suits for each. The formula would then be combin(6.4)*combin(4,1)**5, which is 15,360 winning combinations in the universe of 2.6 million. (The "**" operator means raise to a power.)

The chances of being dealt four cards that match the criterion are 9.46%. The first card, the high card, is pre-specified and can appear in any one of four suits. The remaining three cards must be lower than that card, and will appear in one of six ranks. The last card could be any one of the remaining 48 in the deck, including both high and low cards and without concern if it pairs up with anything. The formula would then be combin(6,3)*combin(4,1)**4*combin(48,1). This yields 245,760 winning combinations.

The chances of being dealt three cards that match the criterion are 41.67%. That's combin(6,2)*combin(4,1)**3*combin(48,2). The high card is once again pre-determined (really, it's a combin(1,1) designation, which is just 1). Of the six possible inferior ranks, two must be selected. Each of the three cards appears in any one of four suits. The last two cards are from combin(48,2) remaining in the unseen deck. So combin(6,2)*combin(4,1)**3*combin(48,2) yields 1,082,880 winning combinations.

Here is a table showing the chances of being dealt four- and three- to a high-hand, by high card. For Ace-to-Five calculations, knock the column heading down by one. Remember, with four- and three-card combos the "other" cards can be anything at all, even pairing up with one of the low cards, or with each other. The chances of getting three cards out of five equal to or lower than a face card, with no restrictions on the other two cards, are nearly 100%. All numbers are percentages.

Moving from Deals to Draws

If an opponent draws one card, you can assume (absent bluffing) that the hand is four-to-a seven-high, or maybe eight-high, possibly (with looser play or short-handed tables) a bit higher to nine-high or ten-high. What are the chances that the next card drawn will be a non-pairing, non-flushing, non-connected low card? (or - for Ace-to-Five Lowball, just a non-pairing low card?)

In Deuce-to-Seven Lowball, a bit depends on whether the cards already in the hand are suited or connected and what their ranks are. In all events, the way to calculate it is to count the "outs" and divide by 47, the number of unseen cards in the deck. It will be assumed (sensibly) that the card sloughed was not an "out."

For example, imagine the four cards remaining of 5-6-7-8 all suited.

In Deuce-to-Seven Lowball, any two or three would do the trick. A four would cause a straight, as would a nine, and a ten is hardly an "out" for Lowball, though a ten-high is capable of winning. So there are eight outs, except that 2 of them are disqualified because they would make flushes. The probability of hand improvement in Deuce-to-Seven Lowball is 6/47 (12.8%) with suited cards in the hand, and 8/47 (17%) if there is an off suit somewhere in the hand.

In Ace-to-Five Lowball, the same hand has 16 outs because aces are low and flushes and straights are no concern. The chance for hand improvement is 16/47 or 34%. In Ace-to-Five Lowball, a player might have considered off-loading the eight and drawing two cards, as an 8-high hand will often lose in that game. However, a two-card draw can be really ruinous, as will be seen momentarily.

A second example is holding 2-4-5-6 and sloughing a Jack, say. Note that these cards are not connected for purposes of outside straights. If playing Deuce-to-Seven Lowball, a three would be disastrous. The deuce is the lowest card in the deck, so there are no "outs" as Aces. The only hope is to have one of the sevens, or maybe an eight. The seven has a 8.5% chance of showing up, and double that for either a seven or an eight.

In Ace-to-Five Lowball, the ace will help the hand, and a three will, too. So without even considering a seven or eight, the chances of hand improvement are 17% (eight "outs").

To illustrate this point, imagine playing Ace-to-Five Lowball, and receiving A-2-3-4-9 in the deal. With a full table, the 9 is iffy as a winner, so you decide to draw one card. What are the chances that you will get back another 9 or worse? The calculation is the same as ever: There are three more 9's and four each of T-K in the unseen deck. That's 19/47 or 40%. Your chances of being no better off and possibly worse off are about 40%, not counting the pairs. If you add the other three of each of the four held cards in the calculation -- to account for the risk of pairing up -- it becomes 31/47 or 66%! The odds are 2:1 against you for hand improvement by sloughing a 9 and drawing just one card!

Now, suppose the hand were A-2-3-9-K. Would you draw a card? Would you draw two? If the player(s) ahead have stood pat, surely the K has to go. In late position with little action, it may make sense to hang in there. But if anyone raises, the K has to go.

Drawing two cards greatly increases the risk of disaster. Consider holding the A-2-3 and sloughing the 9-K. There are then 9 possibilities of getting a pair to one of the three held cards. This is around 20%. There is about a 6% chance of drawing a new pair. There is almost a 75% risk that one of the two cards will be equal to or higher than 9 -- the lower of the two cards sloughed. Indeed, it would be poor strategy to try to "improve" this hand by swapping out both the 9 and the K. You would probably have to pay something to get to the draw, which overwhelmingly disfavors any improvement of the 9. The K, however, is probably a good draw decision.

When it comes to drawing three cards, the probabilities of getting at least one really bad card become very high. The risk of pairing decreases a bit, as one fewer card is being held, but the threat of drawing a new pair goes up a good bit. Try this exercise. Deal three cards at random from a shuffled deck of 52 cards. Do this several times and note how often at least one of the cards is a Jack or higher. If performed a large number of times, the result will tend towards more than nine times out of ten! Though the three-card draw from a deck of 47 is a bit different, this exercise demonstrates clearly why drawing three cards should be avoided. If just one of the three is a Jack or better, your hand is a likely loser, even if no pairs appear. Thus, it was probably a bad decision to bet that hand to the draw to start with.