Category: wisc.edu archive

I’ll be honest, I don’t think that I’ve ever played the dark side in craps. To me, the fun is winning with the whole table, enjoying the celebration as a shooter hits one point, then another, and one more for good measure. And even in losing, there is some camaraderie and bonding—at least we’re all miserable together. It builds character.

However, a user of my craps simulator, Eric Brewer, found some head-scratching results as he compared the win percentages for some dark and light side strategies. As we will see, the chance of walking away from the table is higher than 50% under some dark side strategies. This phenomenon has a few downsides, of course, but this is surprisingly different from most pass-line strategies. In this post, we’ll dive deep into this surprising feature of the don’t pass strategy and discover more unexpected results along the way.

First, Some Craps Background

Craps experts—feel free to skip this section, I’m going to describe the key strategies used in this post to make sure everyone is on the same page. Both the pass line (which I’ve done a detailed analysis on here) and don’t pass bets are two-stage bets that are foundational to the game of Craps. They are almost exact opposites:

• Phase 1 (when the point if OFF)
  • Pass line wins on 7, 11; loses on 2, 3, 12
  • Don’t pass wins on 2, 3; loses on 7, 11; and pushes on 12
  • any other number (4, 5, 6, 8, 9, 10) sets the point
• Phase 2 (when the point is ON)
  • Pass line wins when the point number is rolled; loses on a 7
  • Don’t pass wins on a 7; loses when the point is rolled

The wins and losses are all for even money. However, odds bets can be added in Phase 2 (up to a table maximum), which pay the true odds for a respective bet:

• Placing odds on pass line pays:
  • 6 to 5 when the point is 6 or 8
  • 3 to 2 when the point is 5 or 9
  • 2 to 1 when the point is 4 or 10
• Laying odds on don’t pass pays:
  • 5 to 6 when the point is 6 or 8
  • 2 to 3 when the point is 5 or 9
  • 1 to 2 when the point is 4 or 10

Because the odds bets pay even money, they don’t carry any house edge. The pass line and don’t pass bets have a reasonably small house edge of 1.41% and 1.36% per bet made. Pass line bets have a 2 to 1 advantage of winning in Phase 1 but are at a disadvantage in Phase 2. Because of this, you can’t take a pass line bet down in Phase 2. Don’t pass bets have an advantage in Phase 2, and so casinos will let you take the bet down at any point.

Let’s Test Some Strategies

First, I wanted to get a sense of the difference between the dark side and light side strategies. Eric had done a similar experiment to this as a baseline, which forms the basis of these results. I changed the odds multipliers since I was curious in lower odds tables and adjusted the number of sessions, testing the following strategies:

• Pass line bet
• Pass line bet with 2x odds
• Pass line bet with 345x odds (depending on the point, as is commonly done)
• Pass line bet with 10x odds
• Don’t pass bet
• Don’t pass bet with 2x odds
• Don’t pass bet with 345x odds
• Don’t pass bet with 10x odds

Each strategy ran for 200,000 sessions, with 10 shooters used per session (this is a common way that people budget their betting, and it is easier to count shooters than to count rolls). I assume a $10 table¹ and use a $100K bankroll to ensure that busting won’t be an issue. If there’s a chance that players can bust, the results will be slightly different.

We can understand the range of possible outcomes by looking at the following density plot, which summarizes the 200,000 sessions:

Even without odds, the two bets still can range within a much as a hundred dollars after 10 shooters. Maxing out odds on a 345x table increases the range to nearly $500 on either side. This emphasizes the need for a large bankroll at tables with increasing minimum bet sizes!

But perhaps more interestingly, the dark side and light side bets seem to almost mirror each other in each panel. The pass line has a slightly higher chance of big wins (since the brown line is higher on the far right side), but the don’t pass has a higher chance of more moderate wins. This pattern flips when considering losses: don’t pass has a higher chance for big losses, while the pass line has a higher chance for small losses.

We can take a deeper look through a more traditional boxplot-style visualization:

We let the main box represent the middle 50% of simulations and let the dark bar show the 50% line. This nicely splits an expected session into four regions, each happening 25% of the time: to the left of the main box, in the box but left of the bar, in the box but right of the bar, and right of the main box. We’re all hoping for the sessions to the right of the main box. To explain what happens in the extremes, I use curly braces {,} to denote the region containing 95% of the simulation data. Only 5% of the time would you expect to see a session outside this range.

The boxplots show an interesting trend more succinctly: The don’t pass strategies have winnings over 50% of the time after you add in odds. The dark brown box, representing don’t pass with 345x odds, has a bar above $0, which means that over 50% of the time, you end up with more money than you started! That was shocking to me when Eric contacted me showing this. He wanted to check that there wasn’t a bug in the simulator, and after some further testing, I’m confident the results are a feature of dark side betting.

Why does this happen? I think that after a point is established, the don’t pass bet is likely to win because it will either have 6:5, 3:2, or 2:1 odds in favor of rolling a 7. Placing odds must give you enough to overcome any pre-point losses, where the don’t pass bet is more likely to lose (it’s 2:1 odds that you roll a 7 or 11 loser than a 2, 3, 12 winner). Thus, over 10 sessions, you will likely win on most of the shooters, resulting in an overall win.

Do these results contradict the mantra where “the house always wins”? Not exactly. The house still has an advantage on the don’t pass bets, as we know (it’s 1.36% per bet made). That advantage is marked with the dashed line in the above plot, which we can see is slightly below $0. Even though a don’t pass with odds bettor wins more than 50% of the time on average, when they lose, it has the potential to be very large. So three $40 wins can get erased by a $150 loss, for example. There’s enough of an edge that the casino will win out in the long run with many players.

The table above gives some hard metrics with these strategies, like the precise win chance and expected edge. For both the dark side and light side the chance of winning increases as the amount of odds increases. While the average winnings vary numerically, the values, in reality, should be constant for the light and dark sides. (I would need to run more simulations to see this, especially for 10x odds strategies, which are very volatile; there is extra volatility here since the number of rolls differs in each session.)

A Deeper Look At Win Percentage

Knowing that the dark side strategies can end up with win percentages over 50%, I wanted to dig deeper and see how far I could push it. I suspected that eventually, if someone stayed at the table long enough, the win percentage would have to drop below 50%. To find out, I set up the following simulation experiment:

• Compare 4 strategies:
  • Pass line bet
  • Pass line bet with 345x odds
  • Don’t pass bet
  • Don’t pass bet with 345x odds
• Compare across an increasing number of shooters, from 1 to 100
• 50,000 sessions simulated for each combination
• $15 table, high bankroll

The results are summarized fairly well in the following plots:

Win Percentage

I want to take a quick look at the below plot because it has several surprising and counterintuitive results in it. Each line provided displays the win percentage of the given strategy as we increase the length of the session.

[Side note: win percentage is somewhat tricky to define because of ties. When you walk away with your initial bankroll, some would consider that a win and some a loss—since I want the win percentage + loss percentage to equal 100%, I put half of the ties into each bucket. Counting all ties as wins results in the line given by the upper shaded area, and counting all ties as losses results in the lower shaded area line, in case you want to make those comparisions]

First, look at the green line in the right plot. While the dark side strategy with 345x odds has a decreasing win percentage, it is surprisingly still above 50% even after 75 shooters. A session of 75 shooters is typically between 500 and 700 rolls, or 3-5 hours of play, if not more.

What surprised me more was that for pass line bet with 345x odds, the win percentage first increases as one stays at the table longer. It does flatten out and eventually starts to dip after 100 or so shooters, but it seems extremely counterintuitive that you’re more likely to win if you stay for a few hours than just one.

I’m impressed by how long the strategies using aggressive odds hold on to decent win percentages. We see that not putting down odds results in a win percentage that starts to drop off as the number of shooters increases—by 100 shooters it’s only about 40% for the pass line and 43% for don’t pass.

House Edge

With all this interesting information, it’s important not to forget that the win percentage is only one metric. If we look at the average winnings, i.e. house edge, we can see a familiar pattern:

The longer we stay at the table, the more we lose on average, under all strategies². The lines on the right plot deviate from the expected straight line because the number of simulations is relatively small, but the trend is clear in both. On average, the don’t pass bets have less house edge, which matches the theoretical quantities.

Volatility

Of course, the longer we stay at the table, the wider range of outcomes we would expect to see. Each box below only shows the middle 50% of the data, so half of the time we end out outside this range. It’s clear how much more risk and reward the strategies with odds have, and so taking advantage of the high win percentages will require substantial bankroll. If not properly bankrolled, we’ve seen in past posts that these volatile strategies can end up with high chances of busting, which is never a fun way to leave the table.

Wrap Up

I have to admit that the analysis here makes the dark side strategies in craps way more interesting to me. We have seen that a don’t pass bet with odds can often keep win percentages above 50% while having a lower house edge than a comparable pass line bet. This high win percentage is also surprisingly robust to large sessions. These nice theoretical features still face my two main drawbacks to the dark side betting:

• To have any fun, you must enjoy winning even when everyone else is losing
• Dark side betting doesn’t allow for large wins as well as pass line betting does

The first point will vary from person to person. But the second point is important to elaborate on. Many gamblers play for those elusive, huge wins. Anyone that is pressing bets at the table is hoping for a long roll by a shooter which will make them enough money for a great story. That memory is worth something extra to them, and they don’t mind suffering a few losing sessions to get it.

But maybe taking the opposite approach can also increase your fun. Having more winning sessions can feel great, as you’re less likely to walk away from the table a loser. When you do lose, it could be dramatic with a don’t pass bet, and so you have to figure out if those few large losses are worth the increased number of wins.

I hope in the future to explore these two sides and some of the strategies that can skew things in either direction. Strategies like triple lux (where you power press bets hoping for 3 hits on a given number) seem like they will push even more for big wins, at the expense of very low win chances. On the opposite end, anything with a martingale-type approach (i.e. doubling your bets on a loss to try and recover) will skew the other direction: you can push your win percentage up higher but the losing sessions will be devastating. But having both options is what makes craps such an interesting game in my mind—in no other casino game can you have such an influence on the style of winning and losing that occurs.

Disclosures

This blog post is for informational and entertainment purposes only. Any form of gambling carries an inherent risk. Never gamble with money you cannot afford to lose. Anyone who believes they, or someone they know, may have a gambling addiction, please seek help. National Problem Gambling Helpline 1-800-522-4700 Call Text or Chat NCPGambling.org

Endnotes

1. $10 isn’t a realistic table minimum in all markets these days, but it’s simple to scale up or down. For example, to convert to a $25 table, you can multiply all dollar values by 2.5.↩︎
2. Also, notice how the two figures have the same scale. Some people claim that adding odds to a pass line bet reduces your house edge. While this is true in terms of percentages, it’s not true in dollar terms. I always like to think of odds as “free variance”, since you get to increase your range of outcomes while not adding any house edge.↩︎

Craps friends, it’s been a while. My craps simulator has been collecting dust (figuratively at least) since July 2019, when I looked into some of the best craps strategies on a budget. The global pandemic hasn’t helped with that either, since I’m unlikely to venture into a casino unless I can be elbow-to-elbow with other players. Craps is more fun that way in my opinion.

For this post, I put 5 systems through my simulator and spent some time crunching the numbers. What we’ll see is that each strategy has unique features—though some have significant drawbacks and make it hard to walk away as a winner. There are also a few surprises about strategy evaluation depending on how you budget for the table. Let’s dive in.

The 5 Craps Systems

For completeness, I’ll summarize the 5 systems here briefly. The systems are based on a $5 table (which I realize are hard to find these days) but can be scaled up appropriately for higher table minimums.

Iron Cross

With Iron Cross, you will win some money on each dice roll, except for a seven, once a point is established.

• $5 pass line bet with 2x odds
• $5 field bet (while the point is ON)
• $10 place bet on 5
• $12 place bet on 6 and 8
• If the point is a 5, 6, or 8, skip the place bet on that number

Hammerlock

I think the idea here is to capitalize on medium-length rolls from a shooter. You hope to sneak in a few place bet wins before ultimately hoping for a 7-out.

• $5 pass line bet
• $5 don’t pass bet, with a lay of $30 in odds
• A phased place bet approach (no matter what the point is):
  • Start with $12 place bet each on 6 and 8
  • If one of these bets wins, shift to $22 inside (place the 5, 6, 8, and 9)
  • If one of the inside bets wins, take all place bets down

Risk 12 (A.k.a. A Strategy For Low Rollers)

Like the name suggests, this system only risks $12 on each shooter but offers some potential to make money on a hot roll.

• Before a point is established:
  • $5 pass line
  • $5 field bet
• After a point is established:
  • $6 place bet on 6 and 8, if you have winnings from the pre-point phase:
    • Generally, if the point is 4/9/10, you can take your field winnings plus $2 for the place bets
    • If the point is 5/6/8, you lost in the field, but if you had won on some combinations of earlier rolls (2, 12, 11) you might have enough to add one or both of the place bets.

I omitted the suggestion of risking the extra $2 on some prop bets (i.e. hard 6, hard 8) when you don’t have enough to place them.

Knockout

This one is pretty simple, as described it’s basically a “doey-don’t” before the point and max odds behind the pass line

• $5 pass line
• $5 don’t pass
• 3-4-5x odds behind the pass line bet

Dice Doctor (A.k.a. “The Best” Craps System)

Want to try your luck on the field? That’s essentially the idea for Dice Doctor’s system:

• $10 field bet, with a bet progression if you win.

The progression is $10 ⇨ $20 ⇨ $15 ⇨ $30 ⇨ $25 ⇨ $50 ⇨ $35 ⇨ $70 ⇨ $50 ⇨ $100 ⇨ $75 ⇨ $150 …

You are doubling on every other win, and moving slowly higher otherwise. After a field loss (which happens 55% of the time), you return to $10.

Let’s Simulate

With the systems rigorously described, I was able to pass them through my python simulator to mimic the results of 200,000 sessions at the craps table. It’s not easy to come up with a perfect comparison, so I decided on letting each session finish after 20 shooters had gone at the table. This corresponds to between 133 and 211 rolls 80% of the time but ranged from 68 to 369 rolls across the 200,000 sessions. The average shooter had about 8.5 rolls.

Each session’s craps table has 1 player following each of the systems and follows a few reasonable table assumptions. Since the systems are designed for different budgets, each player has a budget corresponding to what Best Craps Strategy suggested:

These budgets follow a common approach that guarantees enough bankroll for 10 shooters. Thus, the systems with higher bankrolls are the ones making more bets on the table for each shooter. Budgets play a key role in determining which system is right for you, and the 10 shooter measure provides a reasonable comparison.

The Results Are In

200,000 sessions fly by in about 1 hour of CPU time, and we can take a look at the range of outcomes that each system had. The plot below shows the winnings (final cash – initial bankroll) after the 20 shooters. The height gives a sense of the relative chance of each outcome happening for a given system.

A couple of things jump out right away. Most strategies show a spike on the low end of their winnings, which corresponds to “busting out” with no money left. Different strategies have different peak heights, which gives a sense of the chance of busting. With different budgets, these peaks would change, and with an unlimited budget, we wouldn’t see them at all.

The systems also show significant differences in the range of outcomes (statisticians call this variance). The Risk 12system displays the smallest outcome range, and it is well contained between -$120 and $200. Iron Cross and Dice Doctor have a wide range of outcomes, and they even had occasions where they won as much as $2000 (you have to move the plot over to see it).

By the way, the plot above is interactive. You can click on a strategy in the legend to omit it from the comparison. Hover over an outline to remind you which strategy is which. You can also click and drag to zoom in on a region of interest or reset the view with the “home” button at the top.

In More Detail

We can take a more extensive look at each strategy with some hard metrics. I calculate these measures on a $/hour basis by using the rule-of-thumb that 144 rolls roughly correspond to an hour at the craps table, which I’ve used in previous posts.

Average winnings are similar to the house edge. The Low 25%, Mid 50%, and High 75% of winnings give you a sense of what is likely to happen in any given session. 25% of the time you will fall below the Low, 25% of the time you will fall between Low and Mid, another 25% for between Mid and High, and the luckiest 25% of sessions will be above the High.

This table gives some additional context to the plot. We see that Iron Cross and Dice Doctor have a tremendous house edge of $70 per hour. While they also have some of the highest potential winnings, that house edge is way more than I am ever willing to give away on a $5 table. Plus, this assumes the field has a number that pays triple, and it only gets worse if that is not true.

Hammerlock, Risk 12, and Knockout show a much lower house edge, and in-fact the High 75% for Hammerlock and Knockout at pretty close to the Iron Cross and Dice Doctor. This reveals that their upsides are comparable at that level, while you do much better in the average session.

Risk 12, unsurprisingly, has a much narrower range of potential winnings and losses. This is a conservative strategy, which has a time and place for the budget player. However, it’s important to point out that, on average, Risk 12 has higher losses than Hammerlock. So the narrow range comes at some price for the average session.

Looking at the data in a slightly different way, we can see what these metrics look like in $/20 shooters, i.e. $/session. The below table shows those measures, along with the chance of busting and the chance of walking away with any winnings.

Take another look at the difference between the two tables. The ordering in terms of average winnings changed! In all honesty, I am still not 100% sure what causes this. One reason the metrics differ is that each session has a varying number of rolls, and typically longer rolls are per shooter are better for the systems (except Hammerlock). So looking on a per roll basis, and knowing that it corresponds to exactly 20 shooters, provides some information that may bias the first table. Think about the Knockout strategy; if there were many rolls, then I know the shooters probably were making some points, making the player money. I think this second table is more representative in this case because the number of shooters is what each simulation was restricted to.

Are you more likely to join the table for a fixed time or a fixed number of shooters? This may affect how you budget and choose from relevant strategies. For example, the results from this post apply if you are more likely to play with a fixed number of shooters. The data here would suggest that Knockout has the smallest losses during that session, but not necessarily the lowest losses per roll, among the given systems.

Another metric to consider is the chance of walking away with any winnings. This is simply the percentage of sessions where your ending bankroll is larger than your initial bankroll. For Knockout and Hammerlock, this number is relatively high at 44-45%. However, the other strategies range from 34% to 40%. I think this metric is important to factor in because it shows how often you can walk away feeling good with the session.

Hot Table Or Cold Table?

Most craps players intuitively know that in a given session, someone playing the pass line is going to have almost the opposite luck as someone playing the don’t pass. For the most part, one of those players will walk away as a winner and the other a loser, because the payouts are almost inverted (don’t forget the push on 12 for the don’t pass). But if you play the pass line with odds and your friend plays a place 6/8 strategy, how often do you both end up as winners? or losers?

This notion is formalized statistically as correlation, which measures how similar two things are under identical conditions. Since the results of the simulations were all from the same 200,000 table sessions, I’m able to look at the correlation among the 5 systems.

A correlation close to 1 indicates nearly identical winnings—if you won $100, your friend won close to $100. A correlation close to 0 suggests the strategies aren’t similar, and we can’t tell what your friend won based on your winnings. On the other side, correlations can be close to -1 and suggest that if your friend won, you probably lost. For example, the pass line and don’t pass strategies probably have a correlation close to -1.

The plot above shows the correlation between the 5 systems. Take the top row as an example. Hammerlock has a correlation of 1 with itself (this always happens), but it negatively correlated with the other 4 systems. Hammerlock and Knockout have a correlation of -0.67, which likely comes from one relying on winning the pass odds and the other wanting to win the lay odds. Hammerlock does well on a cold table while Knockout does well with a few hot shooters.

Some of the systems show high correlation, such as Iron Cross vs Knockout (correlation = 0.78) and Risk 12 vs Iron Cross(correlation = 0.74). These three strategies all rely on a pass line bet and do poorly when the shooter 7-outs.

Unsurprisingly, the Dice Doctor strategy shows the lowest correlation with any other strategies. The correlation ranges from 0.09 to 0.22 because the field bet is so different from the pass line, don’t pass, and place bets. The strongest correlation is with Iron Cross, which also features frequent field betting.

Takeaways

There’s a lot of information in this post, but I still feel like we have only scratched the surface in analyzing craps systems. The 5 systems described highlight a variety of different bets and offer a little bit for everyone. Here are a few things that I took away from this analysis:

• Systems involving pass line, don’t pass, place 6/8, and odds bets generally show a lower house edge while still offering a range of outcomes.
• You need to tailor a system to your budget, since some can get quite expensive and require a large bankroll (i.e. Iron Cross, Dice Doctor).
• Evaluating systems can depend strongly on whether your budget is on a “per shooter” or a “per time” basis.
• If you enjoy winning while others are winning, you want to pick a strategy that is highly correlated to typical player strategies that “go with the table”.

Disclosures

For transparency, the folks at Best Craps Strategy provided funding for my time to perform this analysis and write the results. I understand that they are genuinely interested in improving their growing site with good craps information, and will use this information to expand their systems page. I don’t make any money if you check out their site. Links from their site have been removed in this updated version.

This blog post is for informational and entertainment purposes only. Any form of gambling carries an inherent risk. Never gamble with money you cannot afford to lose. Anyone who believes they, or someone they know, may have a gambling addiction, please seek help. National Problem Gambling Helpline 1-800-522-4700 Call Text or Chat NCPGambling.org

Table Assumptions

I make the following assumption for the simulations in this post:

• The minimum bet on the table is $5
• 3-4-5x odds are allowed on pass line bets. This means you can put 3x your pass line bet if the point is 4 or 10, 4x if the point is 5 or 9, and 5x if the point is 6 or 8. This also means you can put 6x your don’t pass bet in odds to receive 3-4-5x the bet if you win.
• For field bets, you get paid double on 2 and triple on 12. Some casinos only pay double on 2 and 12, which would increase the house edge for systems using field bets.
• Players strictly stick to their systems!
• For the Dice Doctor system, I won’t continue to increase the progression beyond the $75, $150 portion, the player just continues to go back and forth between $75 and $150 if they keep winning. This happens very rarely, even over 200,000 table sessions.

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.