Calculating Odds on a Mills Dice & Buckly Bones Dice Mac

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Calculating Odds on a Mills Dice & Buckly Bones Dice Mac

Postby Dave » Wed Nov 03, 2004 9:11 pm

The following article was going given to Dick Bueschel back in 1995. He was planning to put it in one of his articles but it never came to be.

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When I first started collecting slot machines, I went over to another collector friend's house who had a Mills Dice and a Buckley Bones. Needless to say I was fascinated by these machines. He explained to me how they worked and that fascinated me more. Unbelievable, I said to myself, what people will do to design a slot machine. From that point on, I knew I had to have those machines in my collection (especially the Mills Dice). I had no idea how hard (and expensive) they would be to acquire. As years went by all I saw was the prices of the Buckley Bones & Bally Reliance go up. I had yet to find a Mills Dice for sale. Since I do not go to a lot of shows, I still really have no idea how rare these machines are. However, I rarely see Bones/Reliances listed for sale in the classifieds of the magazines I subscribe to. I have never seen a Mills Dice listed for sale in the classifieds. My goal was to first acquire a Mills Dice and then get the Bones or Reliance.

Then it happened! Last Spring (1994) I saw a Mills Dice at the Chicago Show. The outside of the machine looked fantastic. I looked at the price tag and shuttered. However, the person who owned the machine was not around. I hung around for several minutes but the owner didn't return. I decided to check out the rest of the show. About 45 minutes later I swung back by the booth (in hopes of talking to the owner). The good news was the owner was back. The bad news was a SOLD sign sitting on top of the Mills Dice. Bummer!! Oh Well, that will teach me to wander around. At least I knew what one Mills Dice went for. I decided that if I found another one for comparable money I would buy it.

Figuring I was not going to come across a Mills Dice for a long time I decided to purchase a Bones from a collector at the Denver show (August 1994). Image
For being a small show (at least compared to Chicago), the Denver show had a good variety of collectibles (quarter Buckley Bones, nickel Buckley Bones, nickel Bally Reliance, Jennings Triplex, couple of Horsehead Bonus, Cherry Front Watling Rol-A-Top, Mills Futurity, to name a few).

Then it happened (again), I got a message from Kenna Joseffy. She was at a show and came across a Mills Dice in excellent condition. Notice, I said message. I was not in when Kenna called and she left a message on my machine. Kenna left the number for the front desk at the show. I called it but they would not page her. I figured I missed another one. Later that afternoon, John (Kenna's husband) called. He told me the machine was still available. I asked him a couple of quick (real quick) questions about the condition of the machine. Knowing that John and Kenna really know slot machines (inside and out) I told John to purchase it for me. Finally!!! I got my Mills Dice and it only took about 8 years to do it!!

About a week later John got back to Denver. I drove over to his house to examine my new treasure.
Image

This machine is unbelievable (I said to myself (again)). John and Kenna told me that a couple of other collectors were already over to look at it. One of them, Ed Zimmerman, called me later that day. He wanted to come over and "see how that thing worked". I invited him over and together we pulled the mech. out of the case and fiddled with it for quite awhile. We then started wondering on what the pay off percentage of the Mills Dice and Buckley Bones is. We noted all the dice combinations on the Mills Dice and Buckley Bones and then went our separate ways to figure out the odds.

The rest of this article is devoted to just that, calculating the player return for the Mills Dice and Buckley Bones (and Bally Reliance).

First a little information on how the Mills Dice operates.

The Mills Dice machine has a large platter where 30 dice sit in a predetermined order.
Image

The machine has a large plate of class just above the dice which make it impossible for them to rotate. When the handle is pulled, the platter spins around. When it stops, a plunger push out two dice. The machine has a rattle in it to simulate the sound of two dice being shaken. The player can only see two dice at a time and never realizes that the dice are not truly being shaken and tossed out.

The following is a list of the dice combinations as they sit on the platter:.

7, 8, 6, 3, 2, 7, 12, 11, 7, 3, 5, 9, 6, 4, 6, 8, 12, 9, 7, 7, 8, 10, 6, 2, 3, 3, 5, 10, 8, 4

The Mills Dice has three possible bets; Field (pays even money), Eleven (pays 16 for 1), and Come (pays even money).

To calculate the player return on Field and Eleven is very easy. From looking at the possible combinations we see that there is only one eleven combination. Therefore, the odds of getting eleven are 1/30 (3.333%). Since the Eleven bet pays 16 for 1, the true return for this bet is (1/30) * 16 which is 53.3333%.

We do the same type of thing for the Field bet. The field wins if the player hits 2, 5, 9, 10, 11, or 12.
There are 11 such combinations on the platter. Therefore the return is (11/30) * 2 which is 73.333%.

The formula for calculating the come bet is a bit more complicated. The come bet wins if the first roll is seven or eleven and loses if the first roll is 2, 3, or 12. If another number comes up (e.g., 5) then the player shoots again and wins if that same number (5) comes up before a 7 does. From looking at the combinations the machine can generate, we find that the combinations 6 and 8 each appear four times and the combinations 4, 5, 9, and 10 each appear twice. To calculate the percent return to the player for 6 and 8 the following formula is used: per_ret = (2 * (odds of winning / (odds of winning + odds of losing))) * 100. The odds of winning (for a 6 and 8 each) are 4/30. The odds of losing (i.e., getting a 7) are 5/30 (because there are five 7's on the platter). We multiply the formula by 2 because that is what the machine pays for a winner (even money). Therefore, the player return for 6 and 8 is (2 * ((4/30)/(4/30 + 5/30))) * 100 which is 88.888%. Doing the same math for 4, 5, 9, and 10 we get a return of 57.142%.

Now that we know the return when 4, 5, 6, 8, 9, 10 are the point, we must also get the odds of hitting a 7 or 11 on the first roll. There are six combinations that yield 7 or 11, therefore the percent return for 7/11 is (6/30) * 2 which is 40%. Now we must figure the odds of hitting each winning or possible winning combination. The formal equation for calculating the total return on the come bet is:
11
Total Return = * ((Odds of getting i on come out roll) * (% return when come out roll is i))
i=2
The breakdown for all come out rolls is as follows:

Come out Odds of getting Return for Total
Roll this roll this roll Return
2 2/30 0% 0.00%
3 4/30 0% 0.00%
4 2/30 57.14% 3.81%
5 2/30 57.14% 3.81%
6 4/30 88.88% 11.85%
7/11 6/30 200% 40.00%
8 4/30 88.88% 11.85%
9 2/30 57.14% 3.81%
10 2/30 57.14% 3.81%
12 2/30 0% 0.00% 30/30 78.94%

Summing the total return gives us 78.94% return for the Come bet on the Mills Dice.

The Buckley Bones operates on a similar principle to that of the Mills Dice. However, instead of using one large platter to hold the dice it uses two drums. The first drum contain 13 sets of dice (26 dice total) and is only used for the come out (first) roll. The second drum also contain 13 sets of dice but is used for the "point" roll.

The same type of math can be used for the Bones. However, a few differences must be noted.

First the Bones uses one set of combinations for the come out (first) roll and a second set of combinations for the point rolls.

Second, the Bones pays a bonus of 100 coins if the player gets four 7/11's combinations is a row.

Third, the amount won for a point of 4, 5, 9, or 10 is eight coins, points of 6 and 8 win two coins.

The following are the dice combinations for the Bones (at least on my Bones):

Come Out Roll: 8, 4, 2, 5, 8, 12, 6, 7, 6, 8, 8, 9, 11 Point Roll: 7, 5, 7, 7 4, 7, 6, 9, 7, 8, 7, 10, 7

The same formula that was used on the Mills Dice was used for the Bones. The following chart shows the returns for each possible dice combination:


Come Hit Return for Total
out roll Percentage this roll Return
2 1/13 00.00% 00.0000%
3 0/13 00.00% 00.0000%
4 1/13 100.00% 07.6923%
5 1/13 100.00% 07.6923%
6 2/13 25.00% 03.8462%
7/11 2/13 200.0% 30.7692%
8 4/13 25.00% 07.6923%
9 1/13 100.00% 07.6923%
10 0/13 100.00% 00.0000%
12 1/13 00.00% 00.0000%
Four 7/11's in a row 16/28,561 10000% 05.6020%
70.9866%

As the chart indicates my Bones has a total return of 70.9866% .

One interesting thing to note is that it is impossible to get a 10 on the come out roll since there are no 10's in the first drum. I wonder if people playing this ever wondered why a 10 was never rolled on the come out!!

One evening I was talking to another collector who has a Bally Reliance. I asked him what the dice combinations were for the come out and point roll. He took his machine apart and gave me the following information (order of dice not necessarily exact (which does not affect odds)):

Come Out Roll: 2, 3, 4, 5, 6, 7, 8, 9, 10 ,11, 12, 6, 8 Point Roll: 7, 5, 8, 7, 4, 7, 6, 9, 7, 8, 7, 10, 6

This particular Reliance pays four coins for a win when the points are 4, 5, 9, and 10 and pays two coins for the points of 6 and 8.

Using the same math described earlier comes up with a player return of 74.47%.

As a final note: In order to do a sanity check on my math, I wrote a computer program to simulate my Bones, the Bally Reliance, and the Mills Dice. I ran the program for a total of 1 million come out rolls.

The result of the computer simulation is as follows:


Mills Dice Field (Math Calculation/ Computer Result): 73.33%/73.17%
Mills Dice Eleven (Math Calculation/ Computer Result): 53.33%/53.36%
Mills Dice Come (Math Calculation/ Computer Result): 78.94%/79.01%
4 Pay Reliance (Math Calculation/ Computer Result): 74.47%/73.96%
8 Pay Bones (Math Calculation/ Computer Result): 70.99%/70.48%

Number of times four 7/11's came up 496 482 in a row.

The difference between the computer simulation and the math was insignificant.


1. Includes getting four 7/11's in a row
2. Includes getting four 7/11's in a row
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Dave
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