Play the practice game below to see how the count for the point changes with each roll. The practice game demonstrates the don’t pass strategy and automatically lays 10x odds when the count is good for the point, based on a 6 roll window. The CSM parameters model a ShuffleMaster 128 using 6 “decks” (312 total cards). Click on the screenshot to play (hit the Auto button to watch it play by itself):
Card craps is identical to the original game, except that the dice rolls are replaced by two cards dealts from a 5-deck constant shuffling machine (CSM). As most experienced gamblers notice immediately, the odds are a little different because of the shoe. As soon as two cards are dealt from the shoe, the distribution for the next roll changes. This can’t be helped, even if the dealer shuffles the muck into the CSM after every roll. Analysis shows that the average 3-4 rolls buffered in the chute of the CSM actually tilts the game in favor of the don’t pass / don’t come player. Even without counting, the 10x odds don’t pass player has a small edge over the house (about +0.9% of the don’t pass bet). A simple card-counting strategy allows the player to vary the don’t pass (or pass) odds per roll, improving the return to about 1.8% of the flat DC bet. This means for a $5 Don’t Pass bet, you’ll make an average of $.09 per game, laying 10x odds when the count is positive. While this is not a lot of money, it’s a lot of fun, and a card-counter’s dream. You can easily count using a simple 3 roll window! The edge for the odds bet itself reaches +0.5% very frequently. It’s very easy and fun to implement advantage play in this game, which is indistinguishable from normal play (changing odds bets on the fly).
This is a winning game for the don’t pass player playing full odds, no thinking required. A smart player can take advantage of the count of the shoe, and lay odds only when the count is good.
This game punishes the players playing full pass odds. While a pass line bet is a normal 1.4% house edge, taking 10x odds increases the house edge to 3.9%. Do not play pass line odds without counting. If you only take 10x odds when the count is good, you can reduce the house edge down to 0.8%. It’s not a bad idea to play the pass line this way, you’ll just find that there aren’t a lot of opportunities to take odds. The best pass-line points are 5/9, and occasionally the 4/6/8/10 counts will get good (these points are biased against you; you need some good cards to take odds). Thus, your variance is reduced in this game compared to real craps, because you only take odds when you’re +EV (on average) with the count.
Besides trying to grind away at the don’t pass with 10x odds (which is a lot of risk for very little), the beauty of this game is that a simple count greatly improves your ability to predict the next roll. Of course by “greatly”, I mean that its statistically significant, and at times even looks prophetic. But overall, you cannot overcome the vig of the place bets. (At times the lay against the 4/10 is profitable, since you can lay $51 to win $24 at Viejas.) Still, people like placing bets, and don’t care about the vig. If this is you, then you should play this game, and watch the cards towards predicting the next roll. For example, a (2,2) (1,2) (1,1) roll sequence will make the 9 easier to hit (1.43:1 instead of the normal 1.5:1), the 10 easier to hit (1.92:1 instead of 2:1), and the 8 easier to hit (1.18:1 instead of 1.2:1). Conversely, you shouldn’t place bets on the 4/5/6 at this time, since they got much harder to hit (2.11:1, 1.54:1, 1.22:1 respectively).
I’ve heard about craps players that devote their lives towards attaining some control over the dice with their throwing technique. God knows how real this effect is, or how much time people put into attempting it. All I know is that a spreadsheet and a computer program will show anyone how cards from recent rolls affect the distribution of the next roll, often significantly. I’ll explain the details of how it all works below.
The Basic Edge
The reason the game is predictable is because the distribution of the next 2-card “roll” out of the CSM is not the standard, symmetrical 2 dice roll distribution normally associated with craps. Instead, the distribution gets skewed to one side (4/5/6) or another (8/9/10), depending on the recent cards out of the CSM. This is easiest seen by considering a simple shoe instead of a CSM, where we can precisely compute the distribution of the next “roll” from the previously dealt cards. A simple spreadsheet easily makes this calculation for us.
If you download the spreadsheet and play around with the rolls, you can quantify the effect of various dealt cards on the distribution. You’ll see that a “low” roll (both cards <= 3) have a +0.25% (quarter percent) effect on the 4/10 odds EV. Similarly, you'll see that "snake-eyes" (1,1) has an equal effect on the 8 point, as does (2,2) affect the 9 point. These sensitivities are summarized in the table below, showing "good", "bad", and "ugly" rolls for a given point. Here, a +4 count equals a +0.25% edge on your Don’t Pass odds.
|Point||Good Rolls||Bad Rolls||Ugly Rolls||Notes|
|4||⚀ ⚀, ⚀ ⚁
⚁ ⚂, ⚂ ⚂
⚃ ⚃, ⚃ ⚄, ⚃ ⚅
⚄ ⚄, ⚄ ⚅, ⚅ ⚅
|“low” rolls are good
“high” rolls are bad
“mixed” rolls are neutral
|5||No ⚄, ⚅
|One ⚄, ⚅
|⚄ ⚄, ⚄ ⚅, ⚅ ⚅
|Six, Five are key cards
There are no neutral rolls
|Six is the key card
There are no neutral rolls
|Ace is the key card
There are no neutral rolls
|9||No ⚀, ⚁
|One ⚀, ⚁
|⚀ ⚀, ⚀ ⚁, ⚁ ⚁
|Ace, Deuce are key cards
There are no neutral rolls
⚃ ⚃, ⚃ ⚄
⚄ ⚅, ⚅ ⚅
⚀ ⚀, ⚀ ⚁, ⚀ ⚂
⚁ ⚁, ⚁ ⚂, ⚂ ⚂
|“high” rolls are good
“low” rolls are bad
“mixed” rolls are neutral
(If your browser doesn’t properly render the Unicode dice characters in the above table, click for a screenshot of the table)
As an example, we use the spreadsheet to graphically show the effect of 5 "high" rolls (6,6) (6,5) (5,6) (4,4) (5,4) out of a new shoe. We enter these cards into the spreadsheet, and plot the distribution of the next roll out of the shoe.
The distribution is visibly skewed, favoring the low side points (4/5/6) since we’ve just used up a bunch of cards that can be used to form 7 and (8/9/10). At this point, any “don’t” odds for 8/9/10 and any “do” odds for 4/5/6 are +EV. This graph clearly shows the effects of previously dealt cards on the distribution. Next, we’ll examine the effect of the CSM on this process.
The Effect of the CSM
If the game were dealt out of a simple shoe, then our count would give us a perfect view on the distribution of the next “roll”. However, it’d be difficult to track the separate counts for each point. The 4/10 points would be easy to track, but you’d need a separate person to track the other 4 points (5, 6, 8, 9). Ideally, you’d know the exact odds for each bet with perfect tracking (e.g., a spreadsheet).
The use of a CSM to deal the cards actually makes counting easier. Often, the casino employees will tell you that the CSM makes the rolls random, so it’s a fair game. That would be true if the cards that comprise the most recent roll had an equal chance of coming out of the CSM on the next roll. Of course, this isn’t the case, because of the buffer of cards in the chute of the CSM. This buffer, or reservoir, is needed so the dealer can always quickly deal another card out of the CSM without waiting for the shuffle mechanism to spit out another card. This reservoir is kept at some minimum depth, so the chute never “goes dry”. So, you can intuitively see how a quick run of favorable rolls can affect the odds of hitting a point. To get an accurate analysis of this effect, you need to model the shuffler, insert it into a game, then find the correlations.
The game at Viejas is dealt out of a CSM, which holds 312 cards (52 sets of “dice”). They use a ShuffleMaster 126 CSM, which I’ve modeled with 40 slots in the shuffling window, and where the muck is shuffled into a random position of a random slot. I set the minimum reservoir depth to 10 cards, and when the reservoir is lower than this threshold, the shuffler drops an entire slot into the chute. I did not place any limitations on the number of cards in a slot. You can see my source code for the shuffler for complete details on how its modeled.
Analysis shows that a simple windowed count with the above roll count values yields an amazing linear correlation to the (dis)advantage on the point. The graph below shows the correlation between a six roll windowed count and the odds EV/roll for each point. The correlation shows almost perfectly straight and parallel lines for each point. These results match the spreadsheet results for a full shoe. A windowed count is different from a running count, in that you only add up the roll values for the last N-rolls out of the CSM. I’ve analyzed various window depths, and the performance is almost equal for 3,4,5, and 6 roll windows. (Of course, the more rolls you can remember, the more accurate your count is. However, as the window widens, the correlation decreases slightly.) Personally, I find it very easy to use a 6-roll window for the 4/10 and 6/8, but I’m lucky if I can remember the last 4 rolls for the 5/9 count.
While the above graph shows the expected return (EV) of an odds bet per roll, I de-normalized the results to show the odds bet “advantage” vs the count below:
The meaning of “odds bet advantage” in the above graph is the EV of the bet excluding non-event rolls (not 7 or point). So when you have a +12 count against a 4 point, you’d make an average 2.5% profit of the entire lay, when you tally up your net results over the long run.
This graph shows that on average, we’re +EV for laying odds against the 4/6/8/10 points when the count is >= 0. The 5/9 points are “fair”, in that it’s odds are the same for a regular craps game when the count is neutral. These are the same biases that we saw in the spreadsheet.
I’ve blogged about various counting strategies (see my card craps posts), but I’ll summarize the basics of practical counting here.
4 and 10 Points
It’s especially easy to count the 4 and 10 points, because these counts are multiplicative inverses (x -1) of each other. I also call this the hi/lo count, since it reflects the net of high (both die >= 4) and low (both die <= 3) rolls in the window. For example, if the hi/lo count is 0 (neutral), then it's to your advantage to lay DP odds against both the 4 and 10 points. Furthermore, if the hi/lo count is neutral on the come out, and the point comes out 4, then you're at a +4 count against the 4, which yields almost a 1.5% advantage for your DP odds. If a high roll (e.g., `Yo) comes out next, this brings the windowed count back to 0, but you still have a 0.5% advantage for your DP odds. If a high roll comes out again within the window, then the count is -4, and you should not lay DP odds against the 4. On the other hand, you'd have almost a 0.5% advantage taking odds on the 4 point.
6 and 8 Points
I think it’s reasonably easy to maintain a 6-roll window count on the 6/8 points. Usually, I’m playing with another counter, so we can ask each other if they remember the key cards (Ace for the 8 point, and 6-spots for the 6 point). In general, its okay to lay against the 6 or 8 if there’s 2 or less key cards in the 6 roll window. Notice that 2 key cards in the window corresponds to a neutral (0) count, but the game is biased against the 6 and 8, so you still have a 1/4% advantage for your DP odds here. At 3 key cards in the window, the count goes to -3, and you should not lay DP odds. Note that two good (+1) rolls neutralize a bad (-2) roll. Note that it takes 5 good rolls in the window to neutralize an ugly (-5) roll. That doesn’t mean you have to wait for 5 good rolls following boxcars to lay odds against the 6 point. But it does mean that the window needs to contain 5 good rolls to neutralize it. So, if the boxcars came after 5 good rolls, it’s still okay to lay against the 6. You should work out other examples with pencil and paper to see how you’ll count in real life.
5 and 9 Points
Here’s my easy way of knowing whether I should lay the 5/9 points. On the come out, I think back if there was an ugly (-4) roll in the last two rolls. This is pretty easy, because those horn rolls and hard-4/10 stick in your head. If there wasn’t an ugly roll, then the worst you can be is at a neutral count (the come out is +2, and two prior bad (-1) rolls) in a 3-rolll window. So I initially lay against the 5/9 unless I remember an ugly roll in the prior two rolls. Then, I apply a running count for each subsequent roll. So, I’m as accurate as my initial count. Very often, the roll sequence makes your decisions obvious, and the count becomes very easy to track after the point.
Ok, now that we know how the roll distribution is biased given our 3, 4, or 5 count window, what’s the best way to play the game? Well, as I’ve said, the only +EV way to do this is to play Don’t Pass, and change the Odds you lay against the point, based on the count. This is called “Variable Don’t Pass”, and is described below. Another way that is +EV is to sit around and wait for good counts against the 4/10, then lay $51 to win $24 (or various multiples of this amount up to $2000). This is actually +EV, but not by much, since the vig is fairly expensive.
Variable Don’t Pass Odds
In this strategy, you place a Don’t Pass bet, and lay odds when the count is good. Keep track of the count for the given point. If the count goes bad, pick up your odds. Ideally, you may change your odds bet on a roll-by-roll basis. Its pretty easy to track the count for a given point, since you focus on the “key” cards (e.g., the Ace is the key card for the 8 point; the Six is the key card for the 6 point; Aces and Deuces are the key cards for the 9 point; Sixes and Fives are the key cards for the 5 point; high and low rolls are important for the 4/10 points). It gets pretty mechanical once you have some practice. To get +EV results, you need to max out to 10x odds for any positive (> 0) count against the 5/9 points, and for any non-negative (>= 0) count against the 4/6/8/10 points. It’s a lot of variance, so you can trade off EV for peace of mind by laying smaller odds.
Personally, I increase my odds lay by 1 unit for each +4 count (+0.25%). I lay a max of 4x odds against the 4/10, and 2x odds against the 5/6/8/9 points. I can’t beat the 1.4% house edge on the flat bet with these small odds.
Changing your don’t pass odds in the middle of the roll is very typical, even in a regular craps game. So, this type of behaviour is not ruled out as card-counting. I’ve been doing it on every roll, and no one cares.
You can also get +EV results by waiting around for the count to get good against the 4 and 10 points. Any time there’s a +12 count (e.g., three “low” rolls and 2 neutral rolls out of 5), you’re +EV to lay $51 to win $24 against the 4/10 points. This comes up frequently enough, but may not last long before the count goes bad again, either by a bad roll occurring, or when the good rolls fall out of the window. Typically, when I do it, I’ll allow the count to go down to +4 or +8 before I take down the lay. The good thing about this is that you get your vig back when you take the bet down. Yay!
There’s a guy at Viejas who just lays against the 4/10, but he doesn’t count cards. He just puts up the bets until they’re resolved, win or lose. Plus, he bets $500 at a shot, or more.
A fun, but very -EV way to employ the count is to make short-term place bets based on the count, on the most heavily-affected number. It’s fun, because fairly often, you’re right with the predictions. It’s expensive, because the vig on place bets is so -EV, no count in the world will make it good. In the PlayCraps format, it’s really easy to change your bets on-the-fly, because you’re very close to the dealer, and its almost like playing a board game together. You could not take your bets up and down like this in a standard craps tub format.
For example, say all of a sudden I see a (1,1) roll, followed by (2,2), then (1,2). These are all good rolls for the 9 point, and the count is +12. The odds against the 9 are down from 1.5:1 to 1.43:1. It’s still -0.4% to place the 9, but I’m really likely to make the bet anyways. I’ll let it go a few rolls, then take it down if it doesn’t hit or lose to the 7.
I’m also likely to place the 6/8 if I see a good roll sequence. For example, (1,1) and (1,2) and (1,1) bring the odds against the 8 down from 1.2:1 to 1.16:1. The EV on the place bet is down to -0.08%. Its worth it to bet, just for the satisfaction of being Nostradamus. If you call enough numbers, and openly count, the other players look at you, then look at the CSM.
I’ve run simulations for all types of counting systems, with various parameters for the maximum muck depth (cards the dealers collect before feeding the CSM) and CSM buffer depth. All results agree with what a basic spreadsheet will show you. For every good roll against a 4/10 point, you pick up +.25% advantage on your odds. For every good roll against a 5/9 point, you pick up .13% advantage (half as good as the 4/10 point) on your odds. And for every good roll against a 6/8, you pick up 0.055% advantage (1/4 as good as the 4/10 point) on your odds.
The CSM is your friend. It makes counting easier, since you only have to remember the texture of the last 6 rolls. If the dealer allows muck to collect before feeding the CSM, widen your window to include the muck cards. When the dealer feeds the CSM after every roll, just keep track of the texture of the last 3-6 rolls. The floor personnel are not worried about counters in a CSM game, because they’re told you can’t count against it. However, with “dice”, each roll changes the distribution of the next roll, so the effect is much greater than in blackjack.
It’s the same any way you cut it. The game is biased against the pass line, and favors the Don’t Pass. While the edge is small, it plays very well, since counting is very easy, and your advantage can increase dramatically in a few rolls. Furthermore, the game is perfect for betting when the count is good, since you can lay up to 10x odds on your Don’t Pass bet when the count is good. And, you can immediately take down your odds if the count goes bad. It’s a counter’s dream.
- All my posts on card craps.
- Practice Viejas Card Craps Game (Play It!)
- Simplified Card Craps Counting
- Card Craps Counting with Pen & Paper Right at the Table!
- Improved CSM Craps Analysis @ Viejas Casino
- PlayCraps™: A Counter’s Dream
- Simulations for PlayCraps™ @ Viejas Casino, CA (old 6-roll window)
- Spreadsheet for roll distribution, showing effect of dealt cards.
- The game is dealt at Viejas Casino, near San Diego, CA
- Official PlayCraps™ website
- The Wizard Of Odds Card Craps page