Posted by Mission146
Jun 05, 2016

Is it Worth Playing?

There was recently a thread on the Forums started by 100xOdds that pertained to an unusual Craps Promotion at the Horseshoe Casino in Baltimore.  Unfortunately, the Promotion in question will likely either be over or almost over by the time this Article is published, however, I am going to take a shot at analyzing this Promotion in the event that it ever happens again to determine whether or not there is an Advantage to be had.  


As posted by 100xOdds:

“Horseshoe has a craps promo this Sat, 12p- 11p.

Throw 3points and get $100 in promo chips.
And At 11p the top 3 players that hit the most # of points get this:

1st: $5000
2nd: $2500
3rd: $1000

Unfortunately, the min bet will probably be $25. (Max odds = 100x)

If I do play I'll probably bring $3000 and bet like this (even if I'm not shooting): 
$25 Passline/$100 odds, place the 6/8 for $25 each

This promo worth it to be there for 11hrs?
Or just leave when/if Im (sic) lucky enough to roll 3points in a roll?”

The way that I understand the Promotion is that the goal is to roll three consecutive Points prior to a seven-out, if a player is successful in doing this, then the player will receive a $100 Promo Chip that effectively has an Expected Value of $49.29 assuming it can only be bet on an Even Money Proposition.  For these purposes, I’m going to assume that the Promotional Chip will be bet on the Pass Line in order to keep things simple.

The first thing that should be pointed out is that the way 100xOdds suggested he would play is far from the Optimal way to play this Promotion.  For example, there is absolutely no reason to make any Place Bets whatsoever as they have no bearing whatsoever on whether or not one gets to shoot the dice.  If there is an advantage to be had in this Promotion, then making Place Bets would accomplish nothing except making unnecessary Negative Expected Value bets which would detract from any positive Expected Value associated with the Promotion.  

With that said, the best possible way to play this Promotion is to just make Pass Line bets which, as 100xOdds seems to indicate, would be at a Table Minimum of $25 and, therefore, would have an Expected Loss of $0.3525 each.  The simplest way to determine whether or not one would be at an Advantage is to take a look at the $49.29 value of the Promotional Chip and determine how many Pass Line bets would have to be made before it became a zero sum Proposition.  To do this, one would simply need to divide $49.29 by the Expected Loss of $0.3525.  

$49.29/0.3525 = 139.8297872 Pass Line Bets

Therefore, the question becomes: Is the player Expected to make three consecutive points prior to a Seven Out in the time it takes for same player to make 139 Pass LIne bets?

What will ultimately have an impact on this is how many Pass Line bets the player must make before being able to shoot the dice.  However, let us assume for a second that the player playing is the only player at the table.  Another thing that we will assume is that a Point is Established because Come Out winners as well as losses to Crap Numbers on the Come Out would not have a bearing on a player’s ability to make three consecutive points.  However, every Pass LIne bet does carry with it a negative Expected Value, therefore, those rolls must be considered in that regard.

On average, 2/3 Come Out Rolls will result in a Point being Established while 1/3 Come Out Rolls will result in an immediate decision.  What we must do is determine the probability of making a Point when it is Established.  

Ultimately, there are 24 possible combinations of Point Numbers with 6/24 of those being either a Four or Ten, 8/24 of those being a Five or Nine and 10/24 of those being either a Six or an Eight.  When any of these points are established, then the only numbers that matter are either the point numbers themselves or the Seven which, of course, terminates the roll and is a loser on the $100 Promotional chip aspect of things as well.  

What we will now look at is the probability of Establishing a Point and Sevening out on any given roll compared to the probability of Establishing a Point and making it on a given roll:

(6/36 * 3/9) + (8/36 * 4/10) + (10/36 * 5/11)  = .27070707

(6/36 * 6/9) + (8/36 * 6/10) + (10/36 * 6/11) =  .395959596

Come Out Decision = .3333333

With that, we now need to look at just the probability of Making a Point when a point is Established compared to Sevening Out as if those are the only two possibilities.

.2707070707/(.2707070707+.395959596) = .406060606

.39595959596/(.2707070707+.395959596) = .593939394

The goal is to Establish and Make three consecutive points prior to a Seven Out, so we can see that the probability of that happening, given that a Point is established, is:

(.406060606)^3 = 0.06695339

This, of course, will also require Points to be Established on Three separate Come Out rolls, and the average number of Come Out rolls before three points is established is:

3/.6666666 = 4.5

As a result, even on a successful attempt, we expect to see 4.5 Come Out Rolls and have made 4.5 total bets before that happens.  In order to fail, we must establish one Point and the overall probability of failure can be expressed as such:

1-.06695339 = .93304661 or 93.304661% after establishing the first point.

However, there are different ways to fail.  We could fail to make the first point Established, the second point Established or the third Point Established and these will all require a different number of Expected Come Out Rolls in order to happen.  

We already know we have a .593939394 probability of failing on the first point established, and the probability of establishing a point on the Come Out Roll is .6666667, therefore we need to look at the average number of rolls this will take to happen.

1/.6666667 = 1.5

As a result, it will take 1.5 Expected Come Out Rolls for us to Establish a Point and immediately fail by Sevening Out.  The overall probability of failure once the initial Point is Established is .593939394, so we have that probability, essentially, of an Expected 1.5 Rolls...this is probably a good time to start a chart:

We can also fail after Establishing two Points, but failing to make the second one.  The probability of such an event, assuming two Points are Established, is:

.406060606 * .593939394 = .24117539025

This, of course, would result in two Come Out rolls being Points Established, which means that it would take:

2/.6666667 = 3

Three Come out Rolls in order for that to happen.

Finally, it is also possible that we will Establish and make Two Points, but then fail on the third.  In this event, there would still be three Points Established, which would be an Expected 4.5 Come Out Rolls.

The probability of such an event, assuming the Points are Established, can be expressed as such:

.406060606 * .593939394 * .406060606 = .09793182511

Results Probaility after point established Expected number of come out rolls
0 Points made, fail .593939394 1.5
1 Point made, fail .24117539025 3
2 Points made, fail .09793182511 4.5
3 Pointes made, success 0.06695339 4.5
Totals ~1 13.5

Of course, it is not enough to simply know how many Come Out Rolls we will have for each particular result, we now have to determine how many rolls we will experience on an average turn of the dice and then determine how many turns with the dice it will take for us to be successful in Establishing and Making the three consecutive points without a Seven Out.

(.593939394*1.5)+(.24117539025*3)+(.09793182511*4.5)+(.06695339*4.5) = 2.35641872975

In other words, we expect to have 2.35641872975 Come Out Rolls when you are shooting.  

You might wonder why the, ‘Probability After Point Established,’ adds up to one for these numbers of Come Out Rolls and you may ask, ‘Is the shooter not going to shoot after Making/Establishing Three Points?’ my answer to that is, for the time being, I am just attempting to determine whether or not the Promo chip, in and of itself, has a positive Expected Value for this Promotion.  If a player is going for the other part of the Promotion, which is the most total points made, then that player is going to want to continue to roll and, therefore, will experience a greater average number of Come Out Rolls than the 2.35641872975 number above.  

However, if you were only playing in order to try to win the Promotional Chips, then the best strategy (since a player can only win once per turn with the dice) would be to Pass the dice to someone else after Making/Establishing three Consecutive Points and then to avoid betting for as long as possible until it is absolutely required to get another turn with the dice.  I am not aware of the Rules at Horseshoe Baltimore, but at many casinos that I have seen, you must have at least one bet during the previous roll (regardless of what that bet is) in order to get the dice or they will be passed to the closest person who was betting after a Seven Out.

With that out of the way, we will now take a look at how many turns with the dice are Expected in order to successfully win a Promo chip.  That would simply be:

1/0.06695339 = 14.9357635215

If we multiply the Expected Number of turns with the dice by the average number of Come Out Rolls per turn, under the parameters of passing the dice after the Third Point is Established/Made, then we arrive at the following:

2.35641872975*14.9357635215 = 35.1959129052

Therefore, a shooter would expect to shoot 35.1959129052 Come Out Rolls for every one time the shooter eventually Establishes/Makes three Consecutive Points before a Seven Out.  

As we determined before, based on a $25 Pass Line bet, a player could lose 139 Pass Line bets before this became a break even proposition.  As a result, if we take a look at the Expected Number of Come Out Rolls per shooter and compare them to the number of rolls one can lose before the game still yields a negative expectation:

139/35.1959129052 = 3.94932219472

In other words, there can be an average of 3.9493 shooters (yourself included) or fewer in order for this Promotion to have a positive expectation right off the top.  

Of course, it is highly unlikely that a player will find tables this close to empty, especially when a Promotion such as this one (that seems more positive than it is) is actually going on.  Furthermore, this Promotion is not worth very much per hour, let’s take a look at the number of shooters and the Expected Loss while waiting for this event to happen for you:

Number of Shooters Expected Come Out Rolls Expected Loss on Come Out Rolls Value of Promo Chip Overall Value
1 35.2 $12.41 $49.29 $36.88
2 70.4 $24.82 $49.29 $24.47
3 105.6 $37.22 $49.29 $12.07
4 140.8 $49.63 $49.29 ($0.33)

When it comes to the value per hour, that also decreases based on the number of players at the Table, not only because more people are shooting, but also because there will be fewer rolls per hour.  In the best case scenario, you will be the only shooter at the Table and, according to this page:


The average number of rolls that are Come Out rolls is 29.6% of all rolls.  You require 35.1959 Come Out Rolls, on average, in order for this to happen.  According to the chart on the same page, a lone player can expect to move along at 249 Rolls per Hour, which is probably feasible if one is only making Pass Line bets.  As a result, of those 249 Rolls per Hour, 

249 * .296 = 73.704

73.704 of those rolls are going to be Come Out Rolls.  What that means is that you will succeed at this 73.704/35.1959 = 2.09411 times per hour for an hourly expectation of about $77.23.  Obviously, this Promotion would be pretty huge if you were able to be the only player at the Table, unfortunately, you are not going to be able to do that.

We will now move on to two players at the Table which, the page suggests, the two players could get 230 Rolls per Hour in there.  Of course, you are only getting half of those rolls, so as a result, you will roll 115 times per hour.  Of those 115 Rolls:

115 * .296 = 34.04

Only 34.04 of those rolls will be Come Out Rolls, you still require an average of 35.1959 of those to succeed, so as a result, you will succeed about .9672 times per hour.  In this case, you would make about $23.67 per hour off of the Promotion.

We will do three shooters and stop there since four shooters seems to have a slightly negative expectation to begin with.  With three shooters, according to that page, we can expect to see 216 Rolls per Hour, but you are only getting a third of those which is 72 rolls per hour.

72 * .296 = 21.312

Only 21.312 of those rolls are going to be Come Out Rolls and you still require 35.1959 Come Out Rolls in order to succeed, therefore, you will succeed about .60553 times per hour which gives you an expectation of winning about $7.31 an hour.

In order to simplify this, a player simply needs to be the shooter a little better than one out of four times in order to have a Positive Expectation.  In order for that to happen, there are a number of tactics the player can employ, such as:

A.)  Switching Tables

-The player could switch Tables as often as possible to get himself/herself closer to the current shooter.  Of course, one could end up effectively blocking oneself out of having a playing spot if all of the tables that are applicable to the Promotion get filled, and if that happens, the player will be shooting zero times per hour.  Of course, shooting zero times per hour (based on the $25 Minimum, of course) is still better than being the shooter only 1 in 3.94 times, or fewer.

There is a second obstacle to this and that is that one must decide whether to play for the overall Promotion or just for the $100 Promotional Chips aspect of it.  If one is playing for the overall Promotion, then it is possible one does not need to shoot quite as often because the overall Promotion will contribute some added Expected Value as it is based on total points made that day.  Of course, the player who wants to better his/her chances will play during the entire eleven hours while a player who just wants to play for the Promotional chips can jump in or out as often as he or she pleases.  

There is also going to be a balance between waiting to find a spot that would make you the, ‘Next Shooter,’ as opposed to waiting if there is going to be a shooter before you.  The reason that is the case is because then a player may find himself waiting too much to get a spot at a table when a player could otherwise be waiting for the current shooter to finish his/her hand as well as one shooter after that person.  While there is a better Expected Value if you are the very next shooter as opposed to waiting for another one, there is absolutely no Expected Value in standing around doing nothing.  Again, though, you still would want to be the shooter no worse than every 1 in 3.94 times to have any kind of advantage on the Promotional Chip aspect of the play.

B.)  Sitting Out (if tolerated)

If the crew will tolerate you not making any bets whatsoever while you are waiting for the dice to come back around to you then, while boring, this increases your Expected Value on the Promotional Chip aspect of the play.  However, this would also require looking at a worst-case scenario which is a completely full Table with ten other players.  The page linked above indicates that there will only be about 102 Rolls Per Hour Expected at that Table.  That is actually pretty awful because you are only getting an average of 9.273 of those rolls of which 2.745 of them are Come Out Rolls.  That means you can be Expected to succeed about 0.077992 times per hour at the rate of $36.88 overall value per success because you are only betting when you are shooting.  However, this only results in $2.88/hour in value and a whole lot of standing around doing nothing.  In fact, if the Promotion runs for eleven hours and the rate of success every hour, you are expected to succeed .857912 times during that entire eleven hours, which is less than once.  That is proven by the fact that your overall success pays $36.88 when you are only betting when you have the dice, but $2.88/hour * 11 = $31.68 which is less than the value of one success and $36.88 * .857912 = $31.64 with differences due to rounding.

Ultimately, while sitting out is a profitable decision, it is going to be extremely boring and also not worth much per hour if there are a large number of people at the table.

C.)  Be Every Player

It is extremely unlikely that you would get away with this for long, if at all, but if you had a team that locked up all the spots on a table and each team member could only bet when he/she has the dice in hand, then every player would be playing as if he/she were the only person.  The table could be sped up a bit by throwing the dice as fast as you get them but there are two problems:

1.)  The casino absolutely would not let you do this. 


2.)  The expected win per hour gets divided up among all of the team members, which is ultimately the same thing as only shooting when you have the dice!

Conclusion on Promotional Chips

When it comes to playing for the Promotional Chips, as a worst-case scenario, you would want to shoot, essentially, one in every three times that you are betting on a shooter and would want to stick to Pass Line bets at the $25 while making no other bets.  One could make an argument for Odds Bets, but I would not recommend those because, while they have neither a positive or negative expectation, they are reducing the Expected Value of your overall action on a percentage basis.  For the reason that players usually take the Odds (because some subscribe to the position that it reduces the House Edge on the overall action) Taking the Odds when you are effectively at an advantage reduces the Player Edge that you enjoy for the same reason.  

Furthermore, even if you shoot one out of every three shooters, you are still not going to quite see that $7.31/hour because that figure is based on only three shooters being at the table and shooting constantly, which is very different from Table Hopping (you’re not playing the entire time because you are switching Tables) from table to table when all of the Tables in question are likely to have more than three players. (the game still moves slower) I’m not even going to attempt to figure out how much one could expect to win per hour by being the Shooter on one out of every three Shooters upon whom the player is making bets, but suffice it to say, the player will make substantially less than that $7.31/hour figure.

A more worthy goal would be to jump in and play when you have the opportunity to be the next shooter, if you are able to do that frequently enough.  Even if you have to throw out a Come Bet in the middle of someone else’s roll, you are still betting in such a fashion that makes you the Shooter every other time, but again, you might be doing a whole lot of nothing before you get the opportunity to jump in and that is going to have an adverse impact on your hourly profit expectation.

Is This a Good Promotion Overall?

Even considering just the Promotional Chips, this is a pretty good Promotion that could certainly be beaten on the basis of Expected Value.  If you are going to be playing Craps anyway, then this is certainly better than sitting there and making Negative Expected Value bets sans any other benefit.  

In terms of Expected Profit per Hour, there probably isn’t going to be much of that, even Table Hopping, simply because of how long you have to wait in order to shoot the dice.  If you could be the only player at the table, then obviously, you would want to play all day and you would make an absolute killing!

That brings us back to this aspect of the Promotion:

And At 11p the top 3 players that hit the most # of points get this:

1st: $5000
2nd: $2500
3rd: $1000

It is really difficult to determine the value of this aspect because you have no idea how many people will plan to shoot the entire time, but there is a pretty good chance that there is a good bit of value to be had here.  Three people will get prizes totalling $8500 which means that the average prize is $2,833.33.  In the event that you are going to be playing the promotion the entire time, assuming that all of the Tables are full (or nearly full) and you would not be strategizing by Table Hopping or Sitting Out, even then at a $25 Minimum you would expect to see 102 rolls per hour over 11 hours for 1,122 total rolls of which .296, or 332.112 would be Come Out Rolls.  These Come Out Rolls have an Expected Loss of $117.07 based on a $25 Pass LIne Bet, therefore, if:

2833.33/117.07 = 24.202

In other words, assuming that none of the players are going to be strategizing in any way by Table Hopping, then if there are less than 24.2 players, you would have enough equity in the drawing that based on that Expected Loss of $117.07 over the 11 hour period that playing straight up would have a positive Expected Value.

If you combine that with Table Hopping as well as Sitting Out when you are nowhere near being the shooter (unless you are admonished for either/both) of those things, then your Expected Value increases in a way that is difficult to mathematically determine without knowing the specific circumstances.  However, your Expected Value is reduced if the Table is moving at a faster number of rolls per hour than that, or obviously, if you are making unnecessary bets.

I suppose the first thing to do would be to try to get a gauge on how many people are going to be playing for the full eleven hours, beyond that, I would also suggest forming a prize sharing partnership (with respect to the big prizes) with anyone who you can rely on to be there for the full eleven hours that way you can come closer to approaching the Expected Value of the play, if indeed it is going to be positive.  

Final Conclusion

If you can form a partnership with people who are going to be there for the entire eleven hours with respect to the big prizes as well as establishing that you will have sufficient equity on the big prizes, then this could be a worthwhile Promotion.  However, it becomes more difficult when more than 24 players are going to be there the entire time, but again, that is assuming that none of them are Table Hopping in a strategic way which you and any partner you have for the big prizes should be doing in order to increase your probability of finishing in the Top Three in Made Points.

With respect to the Promotional Chips, as discussed, at worst you would want to only make bets on the Shooter if you were the Shooter probably every 1 in 3 shooters, or better, to have  reasonably good hourly Expectation on that.  If you are playing in a fashion that makes the Promotional Chips profitable (which also would include being the shooter as often as possible) than any Equity in the final Top Three aspect of the Promotion would be icing on the cake...or perhaps...the Promotional Chips might be the icing on the cake.  That’s really difficult to say without knowing the specific, ‘On the Ground,’ circumstances of the Top Three drawing thing.

In any event, both of these Promotions can be played profitably on an independent or combined basis under certain conditions and/or with certain strategies.  What is a worthy play on an Expected Value per Hour Basis is going to vary from one person to another, so making that determination is also going to be relevant if this is a Promotion that you would even consider playing in the first place.

Even if one does not play this Promotion at an advantage by strategizing, however, any Promotion would still reduce the overall House Edge on the game and is, therefore, better for that player than simply playing straight up if the player would otherwise be doing that, anyway.  

I think the biggest problem that some players would have is to restrict themselves to a Pass Line or Don’t Pass bet only with/without Odds and to avoid Place Bets, Field Bets, Come Bets or worse if those are the sort of bets that make the game of Craps entertaining for that person.  However, the key to taking advantage of Promotions is to determine whether or not the math works out in an advantageous way, and in order for that to most likely be the case for something like this, unnecessary bets that are fundamentally Negative Expectation should be avoided at all costs...even the cost of abject boredom.

Hopefully this Article will be useful for you if you come across a Promotion reasonably close to this, and if not, the concepts in this Article can be applied to other Promotions.  As always, thank you for reading and, please leave a comment!


odiousgambit Jun 06, 2016

wow, you really pored over this! A real White Paper.

The math doesn't seem too tough, but it is sort of 'simple but complicated' where anyone could make a mistake. Did you get anyone to check it?

Mission146 Jun 11, 2016


Thanks for the compliment!

The Math itself was simple in the sense that it was figuring out how to, 'Look at it the right way,' that was difficult. For example, the Math on the Promotional Chip aspect of the thing would have been more difficult if I did not simply assume that the player stopped upon Establishing/Making three consecutive points as that is Optimal as far as the Promotional Chip is concerned. While it would not have made a huge EV difference overall, one would then have to go through and determine how much value one is losing by continuing to roll more than three Points while gaining value on the drawing aspect of the thing by Making more points.

As I alluded to in the Article, the, "Most Points," aspect of it really cannot be figured out because there is too much information missing that would be necessary to coming up with something concrete. It is for that reason that I treated the two separately and why I could come up with something more definitive on the Promotional Chip aspect of the Promotion. Of course, on that aspect, it still comes down to Rolls per Hour (which is based on an average) so that will also have an impact on earnings potential.

It's kind of like looking at how many plays per hour you might get on something like a jackpot promotion: You don't really know because you do not know how long it is going to take for payouts on the jackpots to take place. The best anyone can do is just assume something and go from there, but that's why I showed the math so someone who wanted to operate under different assumptions could do so if he/she wanted.

100xOdds Jun 12, 2016

One thing to add:
you can win the $100 promo chip EACH time you're the shooter.
It's not one per person.

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