House Advantages on Combined Bets
House advantage on the box numbers is a bit tricky, and in fact can be looked at in a number of ways, giving different apparent results. And the house advantage for place number varies continuously depending on how long it remains working.
To begin, let's look at this graph:
This graph depicts the house advantage depending on the length of time a wager is "working" on a box number.
Note that the FIELD advantage is a FLAT LINE, because field bets are decided affirmatively on every roll, thus the expected value for the field bet never changes.
However, the box numbers involve more complicated math, since they are "X before Y" events, where they are not necessarily affirmatively decided at each roll. This also means that if you place a box number, and then take it down after one roll, the house advantage is significantly lower than if you leave it up for many rolls.
The reason is the math for event X occurring before event Y when X+Y is less than the total number of possible events, is:
So where each roll is N, and N is 0 to infinity (event 0 is the probability for the first roll), this is for the 6 or 8:
((Sum of (5/36)*((1-5/36-6/36)^N))*7 + (Sum of (6/36)*((1-6/36-5/36)^N))*-6) / 6
So we can see that when N = 0, the house advantage is much lower than when N = 10 or more.
But what gets weird, and what happens when you combine box numbers together is that there is ambiguity as to how you define a "wager".
And thus, the AMOUNT you APPARENTLY wager can change on a per decision basis, resulting in a skewing of the apparent house advantage.
What I mean is, if you consider an INSIDE bet to be a single wager, then the per-decision house advantage is -1.13636%, but assuming that each box wager is separate, placed at the same time, the house advantage is -2.64463%.
The difference is in how you perceive the "total amount wagered". Do you resolve each box bet individually, or as a group?
And is this the best way to describe box expected value in the first place?
Because place bets on box numbers are not contract bets and can be removed at any time, and the res=olution is ambiguous, and that leads to these ambiguities at to how you look at the EV, perhaps it's better to examine instead the "total amount won" vs the "total amount lost".
If you look at the ration of total amount won (total chips the dealer hands you) vs total amount lost (total amount of chips placed on the table), then we get a figure that is 2.72% for the 6 or 8 and 4.76% for an inside wager. But this is out of line with the "usual" method of determining the house advantage for other wagers. But we do learn something - when looking at place wagers using this method, we lear that the 6 or 8 separately have the same advantage over the long term as the 6 and 8 together.
As such, for the EV for combination wagers to have any meaning or correlation with other, single, wagers, they cannot be combined as a 'single wager" and instead must be combined as a series of independent wagers.
The error in my previous post was that I was combining wagers in a way that skewed the "amount wagered" is such a way that it became less relevant in the context of other single roll bets and contract bets.
Correcting this then, the "in context" house advantage figures are:
6/8: -1.51515% (together or separate) 5/9: -4.00000% 4/10:-6.66663%
Inside: -2.64463% Across:-3.90150%
Iron Cross: -2.45202% (12 is 3X)
Iron Cross: -3.14646% (12 is 2X)*
As you can see, the Iron Cross is actually a bit better than INSIDE when the 12 pays 3 to 1. And still better than ACROSS even when the 12 pays 2 to 1.
This also has implications to proving that ISR is a better strategy than WOTCO, however, there is some more work to show that affirmatively.
Dances With Dice
*(note: there is a possible discrepancy with the Iron Cross results where the 12 is 2X, and I've had some simulations indicate it as high as 3.8%).