Discount Gambling

I’ve been playing more and more card craps @ Viejas, because I’ve found a more socially acceptable and enjoyable way to take advantage of the Don’t Pass bias of the game. Counting is very easy now, since I only need to remember a few things about the last 3 rolls. Ideally, playing 10x odds with the strategy below yields about +1% player advantage of the flat bet (+EV). And whenever you’re laying/taking odds, you have a relatively large advantage on your odds.

Here’s what I do. I start out with a $5 bet on both the Pass Line and the Don’t Pass Line for every hand. Then I either take or lay odds depending on the count for the point. Yes, I know this will cost me more house edge, but there’s a few good reasons to make this additional bet. First, people always look at the “Don’t player” with a little bit of animosity, and being “against” the table takes a lot of fun out of the game. Secondly, you can easily account for how much “house edge” you’ve paid to play, by adding up all the times boxcars hit on the come out roll. Also, the occasional time you lose $5 to boxcars is a lot less aggravating than losing 4 or 5 DP bets in a row to 7-11. But most importantly, if the count gets good during the roll (esp. for the 5/9, and 6/8), you can take pass line odds and cheer with the table for the point. And, when you’re right, you get some extra non-monetary satisfaction.

3-Roll Window Strategy

I’ll review the simple counting strategy for a 3-roll window. You get very little improvement for increasing the window to 4, 5, or 6 rolls, so it’s not worth the considerable effort to try to remember anything more than a few features of the last 3 rolls.

The counts for pass and don’t pass on a given point are multiplicative inverses of each other (i.e., times -1). So while `Yo is +4 for the pass line odds on the 6-point, it is -4 against the don’t pass odds for the 6-point, etc.

When the conditions aren’t good to lay odds, then they’re automatically good to take pass line odds. This is how you play. Start off with both a minimum pass line and don’t pass flat bet. As soon as the point is established, think back about the previous 3 rolls. Lay odds if the conditions are met for the point as described in the above table. Otherwise, it’s good to take odds on the pass line bet. So you’re constantly monitoring the key cards for your point, and adjusting your odds bet accordingly.

Odds Advantage Depends On Count

You can read more about the fair-weighted count values for each roll on my main page on card craps. While the above table works fine for knowing when to lay and when to take odds on a point, knowing the details of the count will tell you how good your odds are. The following graph shows the advantage you obtain on your odds (pass and dont) for a given 3-roll windowed count. (When the count is positive, lay odds; when the count is negative, take odds. You always have an advantage on your odds.)

Don't Pass Advantage for 3 Roll Window

The tables below lists the possible count values and advantage for the points shown in the above graph.

In the above tables and graphs, “odds advantage” means that on average return of your odds bet for a given count, excluding no result rolls. For example, if the count is +12 against a 10 point (the last three rolls were high), then your advantage of 7-out is 3.1% of your total lay bet, compared to hitting another 10. Of course, your actual return is lower, because no result is likely in the next roll. But, you’re (2)(+3.1%) = 6.2% more likely to hit a seven vs. a 10 for a +12 count than for a neutral count.

So overall, you’ll have about a +1% player edge on your flat bet. This, of course, is very little to get excited about on its own. However, you can see that you’ll have a relatively large “advantage” for your odds bet on most rolls. So, when the count is really good, it’s really fun to lay 10x odds, or take 10x odds. For example, say the count is -8 on a 6. It’s a good time to take 10x odds on your pass line bet for the 6. Your chances of hitting the 6, weighted by the 6:5 payout, is 1.1% better than your chances of 7-out. Of course, most likely, the next roll will not be a 6 or a 7, and the count will change. But when you take or lay your odds, you have the edge.

Watching Rolls

After a little practice, it gets easy to count in a 3-roll window. A little discussion with examples will help you get the idea on how to count at the table.

When watching the come out roll, keep in mind the previous roll. Say the come out is (5,2) = 7 pass line winner. You push. Say the next roll is (5,3) = point 8. If you don’t remember snake eyes two rolls ago, then lay some odds against the 8. Now, watch the rolls, and pay attention to the Aces. One ace in the 3 roll window is okay. However, if you see two Aces in the last three rolls, move to the pass line odds. So you’re just paying attention to the aces, and where they are in the window. Aces in consecutive rolls means you’re going to bet pass line odds. If the next roll doesn’t have an Ace, it’s still good for the pass line for one roll. However, you need an Ace to remain on the pass line, otherwise switch and lay odds against. The 6 point is similar to the 8 point, except the six-spot is the key card.

The 4 and 10 points are the easiest to play. It’s really easy to see “high” and “low” rolls. Low rolls are when both dice have 3 or less spots. High rolls are when both dice have 4 or more spots. Mixed rolls are neutral. When a point comes out as 4 or 10, it’s likely that the count is good to lay against. If a 3 neutral rolls go by, then the windowed count is neutral, and still favor laying against the point by +0.4%. However, as soon as the count goes from neutral to negative (e.g., Ace-Deuce vs. 10 point), then take odds on the pass line.

The 5 and 9 points are fun to play, because they’re “fair” and not biased. Of course, the roll establishing the point is +2 against itself, so it’s likely you’ll start out laying against the point. However, the count swings + and – very quickly, and any time there are 2 or more key cards in the window, take odds on the pass line. For example, say the come out roll is (6,5) `Yo. The next roll is (2,3). You should definitely take odds on the pass line, because the count is at least +2, depending on what the roll before the `Yo was. Now, watch the next roll. No matter what, the count will still be at least +2, because of the +4 `Yo, so you always place odds for at least two rolls after a +4 roll.

Conclusion

The game is really fun for counting. It makes the game a lot more fun than watching completely random dice, and relying on pure luck. Card craps has a reasonable component of skill, because odds change significantly with every roll.

Of course, this kind of thing might not be what the typical craps player is looking for. I’ve found that almost everyone tries to play “regular craps” at the card craps table. This can be a problem, because blind pass line odds in this game gives the house a huge house advantage. (Taking 10x odds in a dice craps game makes your overall odds better, because it reduces the proportion of the house edge by the odds factor. However, in card craps, there’s an inherent don’t pass bias, where blind 10x don’t pass odds yields a +1.8% players advantage, but blind 10x pass line odds gives the house a 4% edge vs. the nominal 1.4% pass line cost.)

The card craps players at Viejas definitely lean towards Don’t Pass now, even when I’m not at the table 🙂 The astute players understand that the game is unlike dice craps, and the rolls aren’t quite independent of each other. Yesterday a Don’t player I’ve never seen before started to lecture me on this point before I could tell him I agreed. Then a young couple came and started playing DP and laying odds, like they knew what was going on. When the regulars are playing, at least half the table plays Don’t.

Anyways, I decided to clean up and post my Java source code for card craps, including the CSM model, the roll window, etc. You can download my source code, inspect the models, experiment with the parameters, and verify my results (+1.5% of the flat bet @ 10x Dont’ Pass odds using 3-roll count for the current Viejas shuffler; -3.6% for 10x odds Pass Line player!). I’m posting the source code to show how simple the CSM effect is on the craps game. I.e., given a simple but accurate model of the CSM, the Don’t Pass edge follows:

I wanted to know what kind of variance to expect during my card craps sessions playing 5x Don’t Pass odds with a 3 roll window. So I calculated the cumulative distributions for a few strategies, based on a 100 bet bankroll (e.g., $500 @ $5 min Don’t Pass bet), a +100 bet goal (quit if double up), and a 200 game session limit (might take hours). I used simulations to find the probability density function for a single game (hand) of craps, then convolved it the number of session hands (given bankroll and goal) to get the final session distribution. Finally, I plotted the cumulative distribution function, since it’s easier to read for gambling purposes.

Card Craps Session Outcome Distributions

First, I compared regular dice to card craps, both laying 5x DP odds every roll. As the two curves show, they’re very similar, although the card craps game (magenta, -0.1%) returns better than normal dice (red, -1.36%). In both cases, you have more than a 20% chance of busting out (e.g., -$500 @ $5 pass line bet) or doubling up. (If you closely compare the dice and card curves, you’ll see the cards have almost an equal chance of busting out or doubling up, while the dice show a bias towards busting out.)

Next, I found that card craps performs equally well laying 5x odds with a 3-roll window or a 6-roll window. This is a nice find, since it’s pretty easy to implement the 3-roll windowed count, but practically impossible to count with a 6-roll window. [If you look closely, the 6-roll window (green) performs a little better than the 3-roll window (blue)]. The session distribution function shows about a 12% chance of busting out or doubling up. If you see where the cumulative distributions cross 50%, you’ll see that the card craps games are slightly +EV, and the dice game is of course -EV.

Not surprisingly, playing 5x DP odds with card craps has less variance than actual dice. This results mostly from the lower average bet (4.56 units/roll) for a 3-roll window, compared to dice (6.37 units/roll).

Having calculated this cumulative distribution function, I’ll probably play 5x DP with a 3-roll count, and grind out the player reward points. That’s probably the best reason to play, because you’ll easily make the Elite membership level, with it’s $20/day Free Play award. You’ll also probably make more in reward points (1 point = 1 cent) at the game than in +EV (maybe $0.005/roll). Of course, there’s always the satisfaction of beating the house, whatever that’s worth to you.

Lately, I’ve been drawn back to the card craps (PlayCraps™) game at my local Viejas Casino, mostly because I’m willing to lay more odds now with my newfound bankroll. They’ve changed the constant shuffle machine (CSM) to the ShuffleMaster® 126 model, which has a lot more internal slots than the previous shuffler. I thought this might hurt the don’t pass advantage, but I updated my simulator parameters (now 40 slots, and 312 cards) and the results remained as good as before. Most importantly, I added a MIN_RESERVOIR_DEPTH parameter, which is the minimum number of cards in the chute (reservoir) before the CSM drops another slot. I set it to 5 cards, and experimented with various counting window depths, and max odds to find a strategy I was comfortable with.

I found that a window size between 3 and 6 rolls prior rolls didn’t make much of a difference. This was a nice result, because it’s practically impossible to remember more than two prior rolls, and the management won’t let me record rolls with paper and pen anymore 😦 I also found that laying 5x odds is enough to gain about +0.25% over the house. (10x odds yields an 1.8% edge.) Again, this is nice, because 5x is the limit of my comfort zone. So, I’ll bet $5 on the Don’t Pass line, and use a 3-roll window to determine when to lay 5x odds.

Note that a simple counting scheme emerges from the roll types (“good”, “bad”, “ugly”) and the small window. The 4/10 are very easy to play (i.e., know when to lay DP odds). A neutral count for the 4/10 still yield a +0.15% DP odds edge. If the count gets to +12 (e.g., all three rolls “low”), the DP odds edge averages about +0.5%.

The 5/9 points also work out easily. You need 2 out of the last 3 rolls to be “good”. If there’s one “ugly” roll (-4) and two “good” rolls (+2) in the window, you’re DP odds are neutral. The 5/9 points are “fair” in that a neutral count yields no bias against the point. If all three rolls in the window are “good”, then your +6 count yields about a +0.2% DP odds edge. If you have two “good” rolls (+2), and one “bad” roll (-1) in the window, the +3 count yields about a +0.1% DP odds edge.

Finally, the 6/8 points are very easy to play as well. If two of the last three rolls are “good” (+1) and one is “bad” (-2), the neutral count still yields a DP odds edge of +0.1%. If all three rolls are “good”, the +3 count yields a DP odds edge +0.2%.

You don’t have to worry about being perfectly exact on all your counts. Usually, when I play, I pay attention to how the previous hand ended. That way, I know the roll before the come-out. The important thing is to have an adequate bankroll, and the will to lay against all points when the count is good. You’ll find that things average out well, and a game is enjoyable with enough bankroll and a +0.25% (@ 5x) tailwind.