Discount Gambling

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.