Author: Sean Kent

For those that read my last post, you’ll know that if I’m in a casino (especially on a cruise ship), it’s pretty likely I’ll be playing craps. Well, back in March, I was fortunate enough to be going on another cruise and wanted to know the best strategy for myself at the craps table. If you read any craps book (or amateur blog post), they will tell you that the best bet in craps is the pass line bet with as many odds behind it as the casino will allow. I wanted to use my craps simulator to test this theory with a few other common strategies. To go beyond just looking at the house edge, I programmed the simulator with a realistic scenario: someone (i.e. me) walks up to the table with $200, on a table with minimums of $10, and plays for about an hour. After that hour, and their bets finish out, what is their net winnings?

This scenario brings to light a few really interesting concepts in craps, gambling, and statistics at large. We’ll see that house edge matters, but the risk of each strategy (how much your winnings can vary from session to session), and your chance of losing everything can also be important points to consider when choosing a betting strategy. And at the end, I’ll reveal how I did gambling on the cruise following the lessons learned from this analysis.

Too Many Strategies To Choose From

I couldn’t try every strategy in the book for this analysis. Each strategy is a bit tedious to code and I only had a limited time before the cruise. I was able to test 4 strategies that I believe to be common or interesting. The strategies are as follows:

• Pass line bet followed by 2 come bets (i.e. Three Point Molly with no odds)
• Pass line bet with 2x odds and no other bets
• Pass line bet followed by Placing the 6 and 8 after a point is set
• No pass line bet, but Place 6 and 8 after a point is set, and follow with 2 come bets to try and get on 4 numbers

I chose these strategies because they all include some of the lowest house edge bets (pass line and come bets (0.42% per roll), odds bets (with no house edge), and placing the 6 and 8 (0.46% per roll)). They differ in the combinations of the bets chosen, and some of them have hedging involved (e.g. using a come bet to protect a pass line bet’s loss on the first roll). I didn’t include any “dark side” strategies because for me it’s more fun to win when everyone else is winning, but it would be interesting to consider those in a future analysis.

For the avid craps player, these strategies may seem pretty tame, in that they don’t involve many bets or numbers at any given time. We will see that even these simple betting strategies offer plenty of variation in possible outcomes, and with a limited budget can easily end up with the player losing all of their money.

Show Me The Data

After running 200,000 simulations for each strategy, on a table with $10 minimum bets and a buy-in of $200, throughout 144 rolls (which roughly equates to 1 hour of play at a table with 5 players) the relative chance of each outcome is given in the plot below:

All of the strategies follow a similar pattern: they have a large portion of outcomes near $0, a few outcomes where the player won a lot of money, and a spike near -$200 where the player essentially busts out and looses almost all of their initial outlay.

Where the strategies differ is in how big the bust-out spike is, and how much potential upside the strategy has. There’s a lot of valuable information in this plot, but I think it doesn’t quite tell the whole story. For that, we will need some hard numbers.

The Statistics Nerd Stuff

I described in detail many of the metrics in the below table in my last post. They are fairly self-explanatory but give a good picture of the realistic wins and losses one can expect in a given hour of playing at the table.

Here we see the real trade-off between these strategies. The ones with worse average winnings (high house edge) also tend to have lower risk (both in terms of S.D. of winnings and chance of busting), and vice versa. Let’s also visualize some of these table values on the outcome plot:

The winning strategy in terms of average winnings is no surprise: it’s the Pass line with 2x odds. This is the canonical wisdom of almost every craps book you can find. So, what’s the problem? That strategy also has the largest chance of busting at a whopping 29%. For someone who would very much like to play more than an hour or two of craps over a 7-day cruise, that’s too risky. Off the top of my head, I estimate I’d have less than a 10% chance of walking away with any money after 7 days of playing for an hour each ((1−0.29)^7=0.0909).

One strategy I eliminated right away was the Place 6/8 with 2 come bets strategy. This strategy has a high chance of busting (around 25%) but also is the worst in terms of average winnings per hour (I’d expect to lose about $22 each hour). It seems like hedging the place 6/8 bets with two come bets adds to the house edge, and you need the shooter to get on a hot streak to make any money. This is the worst of both worlds compared to the other strategies.

This left the two strategies that I (mostly) employed on the cruise: Pass line w/ 2 come bets and Pass line with Place 6/8 bets. For me, the average winnings per hour weren’t too outrageous, and the chance of busting was much more palatable. Both of these strategies lend themselves well to playing a lot of craps on my trip, minimizing the chance of losing all my money, and still offering the opportunity to walk away with a decent chunk of change. To me, that’s a win.

From The Computer To Reality

On my trip, I ended up mixing between Pass line w/ 2 come bets and Pass line with Place 6/8 bets strategies. However, I didn’t stick to this 100% of the time. Sometimes I added some odds behind my pass line bet if I was feeling riskier, or sometimes I would take my Place 6/8 bets down after they won twice if I was feeling less risky.

Did my strategy work? The problem in the casino is that you don’t get to see the results of 200,000 casino visits—you only get one shot. I didn’t play craps every day, but most of the days I played, I saw some really cold tables. By the fourth day, I was almost out of money. However, by reducing my chance of busting, I kept my money long enough to play on the latter days, when there were a few hot rollers. By the end of the trip, I had won over $300, although some of that was from the blackjack table. And more importantly, I had a lot of fun!

At the very least, it was an interesting exercise to look at these betting strategies in a realistic simulation. The big points here are that Pass line bets with odds can’t be beaten in terms of the house edge, but may be too risky for someone on a limited budget. In general, strategies that involve hedging hurt your house edge, but for some, the lower risk might be worth it. Generally, variance is a good thing in the casino, because without it you’d never walk away from the table a winner. However, knowing the risk that you’re taking when you walk into the casino is always recommended.

I hope you enjoyed this analysis as much as I did. I may explore other strategies in the future, so if you have some in mind, please reach out. I’d love to hear your thoughts and comments

Not too long ago, on a cruise ship in the Caribbean, I was convinced that Craps is by far the most exciting game in the casino. Maybe it’s the dice constantly flying through the air. Maybe it’s the loud shouts from players and dealers: “Dice out!”, “Six, winner!”, “Odds on my ten!”. Or maybe it’s the camaraderie formed by winning and losing with (almost) everyone at the table. Whatever the case, I had to learn how to play.

After an hour or so of getting advice from my dad and uncle, and watching my money fly on and off the table, I was able to figure the game out for the most part. But after the cruise, I sought out some more information from the plethora of online resources about craps.

Unfortunately, roughly a year later, I’m finding that much of this conventional wisdom just doesn’t add up. And I’d like to change that. Let’s start with the most foundational bet in Craps: the Pass Line bet.

The Pass Line Bet

If you’ve never played craps before, I encourage you to learn the basics with a little Google searching. For those in need of a refresher, the pass line bet takes place in two phases and always pays even money. Phase 1: (Point is OFF) bet wins if 7 or 11 is rolled, bet loses if 2, 3, or 12 is rolled. Any other roll moves the bet to the next phase by setting a point to be the number rolled. Phase 2: (Point is ON) bet wins if the point number is rolled again, and loses if 7 is rolled.

Given the two-phase nature of the pass line bet, calculating the “house edge” can be a little tricky. Furthermore, it shouldn’t be completely clear what I mean by “house edge” anyways. For now, let’s assume that the house edge is the percent of a bet the house expects to win once that bet is made. It turns out that the pass line bet has a house edge of 1.41%. I.e. on a $10 pass line bet, you will on average lose about 14 cents. Of course, you either win $10 or lose $10, which is one reason why thinking in terms of house edge is quite difficult.

The other reason that house edge is tough to think about is that most bets in craps are on the table for multiple dice rolls. A pass line bet could be in play for 1 roll, or 100 rolls (although the latter is unlikely). Should this change how we think about the “house edge”? I think so.

To illustrate my point, let’s consider a more realistic scenario. You walk up to a craps table with $500. There are 4 other players at the table with you making a few bets. You place a pass line bet for $5 down, wait for a win/loss on it, and continue placing $5 pass line bets. What are your winnings after an hour of play? I couldn’t find this answer anywhere, so I made a craps simulator in Python. The results of 100,000 simulations (assuming 144 rolls/hour) are shown below:

So, where in this simulation does that 1.41% house edge show up? Nowhere that I can find. And that’s the problem with measuring craps bets with the way we have defined house edge (and the way it’s defined on just about every craps website you will find). In my search, I only saw one website thinking about this problem, and that was Wizard of Odds. There, they define a house edge in terms of bet made, bet resolved, and per roll, which helps to break down the problem and make it more realistic for a craps session. In theory, if I know there’s a house edge of 0.42% per roll (per Wizard of Odds), and I can estimate about 144 rolls per hour at a table of 5 players (also per Wizard of Odds), then I can expect $5 * 144 * 0.0042 = $3.02 in losses per hour, right?

Well, yes and no. In the above simulation, the actual average loss per hour was $2.32. This would correspond to a house edge of 0.32% per roll. So where’s the discrepancy? One small source of discrepancy is the fact that “only” 100,000 simulations were run. Believe it or not, we really need more like 1,000,000 simulations (that’s 144,000,000 dice rolls) for results up to that accuracy. The other, more significant, source of discrepancy is that in my simulations, most of these sessions end with a bet left on the table. It’s uncommon that on the 144th roll, the pass line bet wins or loses. Further, the pass line bet is more likely to win during the first phase, but more likely to be on the table during the second phase, so it’s no surprise my simulation understates the expected losses.

To confirm my suspicions, I ran 1,000,000 simulations of sessions that end at 144 rolls or when all bets are resolved, whichever comes later. After dividing the average loss by the number of rolls and the bet amount, the 0.42% house edge per roll shows up perfectly.

To summarize these results, defining the house edge in craps is hard. I’d say that 1.41% is not the wrong answer, but it is a bad one. For me, the average loss per hour is much better, and much more useful when I’m playing. However, this needs to account for bets still on the table. Further, since most players have the pass line bet as the starting point of their betting strategy, it helps to think of this bet as your “cost to play”, and at $2.32 an hour on a $5 table, that’s not too bad. (Side note, you can easily convert this to whatever table minimum you play at, a $10 table would cost $4.63 an hour). For completeness, let me mention a few more useful metrics for craps betting. I plan to explore more complicated strategies in terms of these metrics.

Better Metrics

Typical winnings refer to the median result, or what happened to the 50th percentile person when sorting the winnings. SD is the standard deviation of winnings, which is important for understanding the risk or variance of a bet. Risk is a double edge sword. Without it, you’re almost guaranteed to lose. With it, you have a chance at coming out ahead, but your potential losses could be magnified.

Low 25% and High 75% refer to the 25th and 75th quantiles of the winnings. A nice interpretation is that you have roughly a 25% chance of leaving after an hour with less than $-25, a 25% chance of leaving with more than $20, and a 50% chance of being somewhere in the middle. For the average gambler, understanding this range for a strategy is (in my opinion) more useful than knowing the house edge for a bet.

All of these metrics should scale up with your bet too. So at a $10 table, simply multiply these numbers by 2, at a $15 table, multiply by 3, etc. That’s just one more perk of thinking in terms of $/hour.

Take Home Message

The house edge is hard to understand in craps, and even harder to define. No matter how you split it, it’s hard to figure out exactly how much you expect to lose without running simulations. Personally, I think losses measured in $/hour are most useful. After all, who counts the number of rolls at the table? But, unless you plan on playing craps for 100,000 rolls, the house edge isn’t going to matter much to you anyways. I’d encourage any serious gambler to think in terms of a range of outcomes, especially low 25% and high 75%. Knowing that half the time I’ll be somewhere between $-25 and $20 after an hour, I’m more than willing to make a pass line bet and partake in the excitement at the craps table.

Want to know how good your craps betting system is? Stay tuned for more posts, as I plan to detail popular strategies in terms of these metrics and compare them based on how the dice shake out. If you can’t wait for those posts, reach out on twitter @Sean__Kent or via email spkent@wisc.edu. I’d love to hear what you want to see next and any feedback you have on this post.

Next week, I plan to discuss all of the misconceptions of the odds bet in craps. Everyone seems to have very strong opinions on these, and we’ll see how they shake out in the simulations.

BONUS: Detailed Pass Line House Edge Calculation

House edge per bet made

To find the house edge per bet made, we need to consider Win/Loss %∗𝑃(outcome) for all possible outcomes. Note: 𝑃(outcome) denotes the probability of some outcome occuring. Let 𝑊 be the event of a win (which pays 100% of the bet) and 𝐿 be the event of a loss (which pays -100% of the bet, i.e. a loss of 100%). Since we have two phases, this turns out to be

Being a little more specific in the possible outcomes for Phase II, we get 

We note that a Win or Loss in Phase II depends on what the point is. In general, 𝑃(Point is 4 or 10)=(3+3)/36=6/36, 𝑃(Point is 5 or 9)=(4+4)/36=8/36, 𝑃(Point is 6 or 8)=(5+5)/36=10/36. Further, we can figure out the Win probability in Phase II by considering outcomes that win over outcomes that either win or lose. That is, 

Similarly, 

All together, the house edge is 

And there you have it, a house edge of 1.41%. There’s a reason this math is not trivial–casinos do not want you to think about these things and know their edge. Even more reason to just run simulations…

Average Passline Rolls

To calculate house edge per roll, we first need to find out how many rolls a pass line bet is on the table for (on average). Fair warning, this includes a lot of math and statistics.

The first step is to break the problem down into two phases, as we have done before. Once a pass line bet is placed, that bet wins and loses in Phase I if a 2, 3, 7, 11, or 12 is rolled, which happens (1+2+6+2+1)/36=12/36 of the time. In that case, we observe 1 roll. Otherwise, a point is set.

Let’s say that the point is a 6. Now, in Phase II, we will either win, lose, or keep rolling. The chance we win or lose is 𝑃(6)+𝑃(7)=(5+6)/36=11/36. Otherwise we keep rolling, adding to the count. In the real world, this phenomenon is quite common. There’s some probability that we end our streak, and a complementary probability that the streak keeps going. For example, think about the number of lottery tickets you buy until you win, the number of apples you purchase until you find one with a worm in it, or the number of emails you have to sift through to find something that’s not spam. To a statistician, these are all instances of the same probabilistic phenomena, called a Geometric Random Variable.

The good news is that this is a well studied phenomena, and statistians even have an expression for the average number of such events until we observe the end of the streak. They found the average to be 1/𝑝, where 𝑝 is the probability the streak ends. In our case, the streak ends with a probability of 11/36, so the expected rolls after the point is 6 is 36/11≈3.273. All we have to do now is consider the other point numbers and the probabilities that they happen.

Last step: put it all together. We multiply the probability of an outcome by the expected number of rolls for that outcome. This involves taking the sum over all values in the above table, where we multiply the second column by the last column + 1. Adding 1 comes from the fact that it took 1 roll to establish the point. In a formula, it looks like this:

A little bit of algebra gives us a final answer of 3.3758. Thus, the house edge per roll should be 1.41% / 3.3758 = 0.4189%, as we saw previously.