Now that Triple Draw Lowball is catching on in popularity in many cardrooms around the country, I thought it would be appropriate to show how to calculate the chances of a common scenario. Playing Deuce-to-Seven Triple Draw Lowball, what are the odds of making a seven low (when the smoke clears) if you start off by drawing two cards to a seven, a deuce, and another low card? (Remember that aces, straights, and flushes count against you. Thus the final answer assumes the first draw is to three cards that can’t possibly turn into a straight or a flush.)
The following calculation assumes that you will always discard any card that does not help you make that seven. In real life you will often keep an eight or sometimes even a nine. Thus in real life, making a seven is rarer than what we will come up with here.
Say you’re dealt A
K
7
3
2
You discard the ace, king and keep the seven, trey, deuce. There are six ways you can wind up with a seven low after three opportunities to draw.
- You can make it immediately.
- You can catch two bad ones on your first draw and then make it on your second draw.
- You can catch two bad ones on both your first and second draw, and then make it on your third draw.
- You can catch two bad ones, then catch one good one, and then make your one card draw.
- You can catch one good one, then miss your second draw, and then make your third.
- You can catch one good one and then make it on your second draw.
Because each of these scenarios is “mutually exclusive,” we need only to figure out the chances of each scenario and add them up. In all cases we will make no assumptions about the cards in the other players’ hands.
The chances of making the seven low immediately is an easy thing to figure. Just multiply 12/47 times 8/46. See why?
With 47 unseen cards there is a 12/47 chance the first card you draw will be good (four, five, or six). If it is, there is an 8/46 chance the next one will be. (Note: It’s not 11/46 because we can’t pair.)
(12/47) (8/46) = 96/2,162 = 4.44%
Scenario Number 2 requires us to first pitch two bad ones, (35/47) (34/46), and then pick two good ones, (12/46) (8/44). Multiplying those fractions gives us 2.56 percent.
The third scenario has us missing twice and then making the last two card draw. Hopefully you see that this is (35/47) (34/46) (33/45) (32/44) (12/43) (8/42). This comes out to 1.56 percent.
In Scenario Four, you again catch two bad ones, (35/47) (34/46), but then you go on to catch one good one on your second draw. So what is the chance of catching enough to give you a one card third draw? Actually, there are three ways that can happen. You can catch a good one (a four, five, or six) followed by a non four, five, or six. You can catch a non four, five, or six followed by a good one. And you can catch a good one followed by a second card of that rank (e.g. four-four, five-five, six-six).
Little-big is (12/45) (33/44) = 396/1,980. Big-little is (33/45) (12/44) = (396/1,980). Little followed by pair is (12/45) (3/44) = 36/1,980. Altogether, the chance of catching one good card on the second draw (after missing your first draw) is 828/1,980. So the chances of missing the first draw and making half the second draw is (35/47) (34/46) (828/1,980), which is 985,320/4,280,760. That must be multiplied by the chances we make the hand on our third (one card) draw: (8/43). Thus Scenario Four occurs
4.28 percent of the time.
Scenario Five has you catching one good one on your first draw, (12/47) (35/46) + (35/47) (12/46) + (12/47) (3/46) or 876/2,162. Then you miss your second draw, 37/45. Then you make it on your third draw, 8/44.
(876/2,162) (37/45) (8/44) = 6.06%
In the last scenario, you once again catch a good one immediately (again 876/2,162 chance) but now hit your hand on the second draw (8/45).
(876/2,162) (8/44) = 7.20%
Altogether we have
Scenario 1: 4.44 percent
Scenario 2: 2.67 percent
Scenario 3: 1.56 percent
Scenario 4: 4.28 percent
Scenario 5: 6.06 percent
Scenario 6: 7.20 percent
That adds up to about 26.2 percent. Again in real life you are more than a 3-to-1 shot because this calculation assumes you will always discard any eight or nine you catch. It also assumes that you will not give up after catching four bad cards. Thus the real life answer is more like 5-to-1.
In any case, the main reason for this article was not to help you to play Triple Draw Lowball, but rather to show how seemingly difficult probability problems often really aren’t. You would be doing yourself a favor if you carefully followed my reasoning.
