Discount Gambling

At some California casinos, craps is dealt using two 6-deck shoes, one for each die in a roll. Of course, the rolls in this type of game are not independent of each other, as a simple spreadsheet will show you. What is surprising is that a simple count of the 6-spot cards will yield +EV opportunities in the Field, provided the casino pays triple (3x) on boxcars, and double (2x) on snake-eyes on this bet. I’ve analyzed the Field bet for this game, assuming the house deals 1/2 the shoe before shuffling. The graph below shows the theoretical distribution of Field Bet expectation values (EVs) for this two shoe craps game.

The above graph shows that with exact knowledge of the cards dealt from the shoes, the Field Bet becomes advantageous (+EV) 6.44% of the time. The average advantage of a +EV Field bet using a perfect count (e.g., a spreadsheet) is 1.2%. If the house deals deeper than 50% of the shoe, these results will improve.

I found that a simple running count of the Six-spot cards is fairly good at extracting the edge out of the Field. The running Six-spot count works as follows:

• Count the number of 6-spot cards contained in every three rolls.
• Every third roll, add (1 – three_roll_six_count) to the running count.
• The Field Bet is +EV when the running count is >= 7.

Intuitively, this count make sense, because you expect to see one 6-spot card in three rolls. The running count reflects the “excess” 6-spots in the decks, i.e., how loaded the decks are with 6-spots. A simple spreadsheet shows that the sensitivity of the 6-spot card is huge on the Field, because it pays 3x for boxcars (6,6). The effect of the Aces is very small compared to the 6-spot card.

The relationship between this simple running 6-count and the Field Bet EV was verified by Monte Carlo analysis, as shown in the following results:

The simple count only yields about 60% of the opportunities found by a perfect count (the count is +EV only 3.8% of the time, compared to the theoretical 6.44% limit). So while this is an interesting find, it’d be a little boring to stand around waiting to bet the Field once every 25 rolls or so. In practice, you’d probably wouldn’t even make a bet for 2 out of 3 shoes, but you’d bet often once the shoe went +EV. Overall, the average advantage per Field bet made is a little higher than 1%, and on average you’d make a net profit of 0.058 Field bets per shoe. So, even if you’re betting $100 on the Field when it goes +EV, you’d only make $5.80 per shoe. That’s a lot of standing around for sub-minimum wage with a lot of risk. But, if you wanted to “take a shot” at making a big win with a small number of big bets, this might just be the ticket for you (especially when the count gets really good).

Often when I play card craps @ my local Viejas Casino, I wonder how profitable it would be to play the Don’t Pass Line, with 10x odds when the count is good. I know I could play lightning fast with a simple 3-roll window for the 5/9 and 6/8 points, and a 6-roll window for the 4/10 points. I know that this strategy would yield an average (+2.2%)(flat bet) per hand, or ($5)(2.2%) = $0.11/hand profit rate. During the week, especially during certain hours, you could be heads up with the dealer, and perhaps average 15 rolls per minute, or 3 hands per minute. So, on average, you could make about $20/hr. More importantly, you’d earn player rewards cash back at a pretty decent rate too. (I’ll take a rough guess that you’d make about $3/hr in cash back reward points.) Additionally, you’d make the MVP player level, which earns an automatic $30/day in cash back. So, if you play this game 8 hours a day, you’d average something on the order of (8hr/day)($20/hr + $3/hr) + $30/day = $214/day.

Ok, sounds great. So why don’t I do it? All I’d have to do is lay $100 against the 4 and 10 points, $75 against the 5/9, and $60 against the 6/8, unless the count is bad. I see people betting this kind of money all the time. It’s not unreasonable.

So I looked into the session outcome distributions for various bankroll, goal, and time-limit scenarios for this +EV game, to see if I could reasonably beat it.

First, I looked at a bankroll of 1000 flat bets (i.e., $5000). I calculated the session outcome distributions, assuming that I quit if I busted out or doubled up (won $5000). I plotted the results as a cumulative distribution function, which is easier to read. The plot below shows the probabilities for both a 1000 hand session (6 to 8 hours), and a 10,000 hand “session” (less than a week).

Cumulative Distribution Function of Card Craps Outcomes @ Viejas (1000 Flat Bet Bankroll).

What I saw immediately from the 10k hand curve (red), is that while I’d have a 33% chance of winning $5000 within a week, I’d actually have an 18% chance of busting out completely. At first, I thought that number was too high, but after I thought about it more, a 1000 flat bet bankroll is only 50 10x lays against the 4/10, or 67 lays against the 5/9, or 83 10x lays against the 6/8. So I can see how everything going wrong (which happens sometimes) could bust you out.

I included a shorter 1000 game session distribution (green) curve for people more likely to try the game for a day. You’ll average a 22 flat bet win ($110 for a $5 flat bet), and 53% of the time your outcome will be +/- 200 flat bets (+/- $1000).

I wondered if doubling the bankroll to 2000 flat bets (i.e., $10,000) would be enough to make busting out nearly impossible. I ran the analysis for a maximum of 25,000 hands, which would take between 3-4 weeks of full time play. The below graph shows that I’d still have a 7.5% chance of busting out and losing my $10,000 bankroll. That’s an improvement from the $5000 bankroll shown (green; 33% chance of busting out), but the risk of busting out is still relatively high. While I’d have a 66% chance of winning overall with the various profits listed, it doesn’t seem worth the risk to me. I don’t think it would appeal to many people. (The $1000 or so in player cash back rewards is not included in this analysis.)

Outcome Probabilities of 1 Month Play 10x Don't Pass w/ Count.

The problem is that while you have a +2.2% EV/game, that percentage is only on the flat bet, and not the odds. You have to lay $100, $75, or $60 to win that $0.11/hand average.