Discount Gambling

Practice Game

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 counts for all the points are displayed, so you can see when place and lay bets improve and worsen. Click on the screenshot to play. You must have Java 1.6 installed on your computer (check your version).

Card craps practice game.

Overview

PlayCraps™ is a sit-down version of craps that’s 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 count with pen and paper right at the table! 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. Your best 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.

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 mitigating 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’s also going to be a nightmare to track the separate counts for each point. (I’m told there is such a single shoe-based game out there; the player didn’t have much luck cracking it, probably because of the difficulty in maintaining all the counts.)

The game at Viejas is dealt out of a CSM, which holds 264 cards (44 sets of “dice”). The reason we can still apply counting to the CSM is that a small number of cards are buffered at the front of the CSM, so the dealer can quickly deal out a number of cards without having to wait for the mechanism. In fact, the cards in the CSM are arranged into 18 slots, where the muck is shuffled into a random position of a random slot. The CSM then drops a random slot down into the buffer for the dealer to draw from, until the buffer empties, at which time another random slot of cards is dropped into the buffer. The slot can contain anywhere from 0 to 20-something cards.

So, some thought and computer modeling of the CSM shows we could calculate the exact distribution of the next roll, provided we know when the slots are dropped. But of course, there’s an opaque cover to the CSM, and we don’t know anything about where we are in the current slot. But, after experimenting with our models, we find we’re not totally out of luck. In fact, just using a simple “windowing” of our count over the last 4 or 5 rolls still produces an amazing correlation between the count and our Don’t Pass Odds EV. The graph below shows this correlation for simulations over a very large number of games.

CSM Craps Counting Advantage

This graph shows that on average, we’re +EV for laying odds against the 4/6/8/10 points when the count is >= 0. On average, we can expect about a +0.40% EV edge on these odds lay for a +8 count. Now, the spreadsheet for a shoe predicts a +0.5% EV edge for the same count, but we’re losing some of the effect due to the uncertainty of the slot in the CSM. For example, we could have seen a (5,4) followed by a (6,6) roll, and laid 10x odds against the 10 point, expecting about a 0.4% to 0.5% edge. However, at this point a new slot could then drop, with a unbiased probability of having the just-shuffled rolls in it. So, we actually have no edge, except the inherent +0.13% edge due to the basic card distribution. So we’re just getting a fair bet, instead of one stacked in our favor.

Note that as we expected, we have no inherent edge on the 5/9 points. We need some rolls to swing the distribution into our favor before laying any odds. Also note that we can’t ever really get a very positive count for the 6/8 points, and we essentially max out at a +4 count. In a CSM, a +4 count is only worth about +0.2% edge, but its better than nothing. Plus, in order to overcome the -1.4% edge of the Don’t Pass flat bet, we need to push our odds. But, as you’d expect, we get a lot of milage from our 4/10 points, because they often swing +8, +12, up to +16 (all four rolls in the window are good). It’s easy to lay odds on these counts, and even more unlikely than usual that the point will hit (e.g., +16 is almost 2.15:1 to hit a 7 before the 4/10 point).

Don’t forget that this is a windowed counting scheme. You only use the last 4 or 5 rolls to compute the count. Anything that happened earlier has a stronger and stronger possibility of being in the slot that you’re dealing out of! If we knew we’re still in the same slot, and the shuffle can’t get to us, then great. But we don’t know. We just have to go with the averages, and limit our “knowledge” to the last 4 rolls.

Various Strategies

Ok, now that we know how the roll distribution is biased given our 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.

Lay No-4/10

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.

Predict-a-Roll

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.

Conclusions

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 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.

Resources

• 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