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From across the pond, never seen anything like this before !!! A Mills Futurity
http://cgi.ebay.com/mills-futurity-anti ... 1|294%3A50

I've seen a couple of Futurities recently, at live auctions where I could touch and prod. It is indeed an ingenious mechanism. If you go ten pulls without a winner, it gives you ten coins back. But this hardly ever happens...

It's a 20 stop mechanism, but with 10 stop starwheels, so you can only ever hit half the symbols. Normally these give an extremely conservative game (30%). However, on the 4th and 10th games the frame holding stop arms moves down half a position, so on those games you hit the alternate set of symbols (200%) that have very many cherries and a better than 75% chance of hitting a '2' payoff. It's probably where they got the idea for those bloody British pub machines where the bonus game gets stuck just under the jackpot position for about ten pounds worth of plays, then you hit a 20p winner and it resets.

Pace made their own version of this, called the Kitty. It had a small window on the front counting the losing plays, instead of a big dial. There was one in last year's spring VCA auction.

Excellent info OB, thanks for this !!

There is another Mills cheating bandit - a small number of Horsehead Bonus machines had a similar layout to the Futurity making it hard to hit a bonus letter.

Right !! .... and wrong !!
You can see the probabilities of this machine (first model of Mills Horse Head Bonus) on my web page : http://www.joiret.com/slot/Mills/bonus/proba_BONUS.php

The Mills Futurity certainly would seem to qualify as a rigged slot machine.
Usually, it works as a 10-stop machine, paying out only about 20% of the money that comes in. On two of the ten spins as the meter moves towards refunding the player's money, however, it gives back twice as much money as is played... mostly in prizes less than ten coins.
Yet despite that, if you include that 10-coin return at the end of the sequence, its payout percentage is actually not that bad.
At least I think so: in my page on slot machines, http://www.quadibloc.com/math/sloint.htm I have attempted to calculate the winning percentage for both the basic Mills Futurity and the Mills Gold Award Futurity.

There is one thing that confuses this issue... Mills made a set of cams for the 10 stop (20 symbol) machines that the operator could change to adjust the payouts, probably used in much the same way as modern slots are key operated to offer bigger payouts on certain days (never play slots on cruise ships other than on the first and second days and the last day ), although in those days the change was most likely made every few months, not every few days.

Incidentally, although I had previously seen the French site that documented the Mills Bonus machine, the URL in the post above mine for it is no longer valid. I've added, therefore, more detail to what was a brief description of the Mills Bonus to my own page.

And now I've updated the page further to add what you've told me about the cams.

Thanks for the explanation of the early BONUS 10/20 system.
I knew early ones had the dual mech., but I never knew why.
I had come to the conclusion that they were simply using-up existing FUTURITY mechs..

I see it wasn't mentioned in this thread yet: the reason for the dual 10-stop arrangement on the early Mills BONUS machines was simply so that players would have to wait 10 spins before getting BONUS at least, instead of getting it in five spins if they were very lucky. However, since the chance of getting a letter from the set B-O-N-U-S was changed from 25% in every spin to 50% in half the spins, and zero in the other half, the long term average chance of getting a letter wasn't changed.

Since the chance of getting the five letters in order in just five spins was very small, either Mills realized the risk of being considered to be cheating wasn't worth it, or it was just because the fancy mechanism was a waste of money.

But if other cams, that change the ratio between the two groups of ten stops, can be put in, since the stops without the letters also don't pay as much back, that would make the machine's percentage easily adjustible. It would be the only slot machine for which the myth of a screw for adjusting the percentage was close to true.

If you really want to delve into the performance statistics you need to track down copies of "Loose Change" mag (January and April 1980 issues). They do come up on ebay and there is the digital archive available at a price. The two parts of this article provided for the first time a detailed analysis of performance and odds of the Futurity by using extensive computer runs taking into account the various random features. Interestingly, it would have been impossible for the designers of the original system to have done anything like these tests. Their instincts alone told them them the machine would perform in the operator's favour. Even the computer tests were not able to come up with a 100% complete statistical analysis.

I have a little web page about slot machines, at this page:
http://www.quadibloc.com/math/sloint.htm
On that page, I work out the house percentage for the regular Futurity (16.01%) and the Futurity Gold Award (27.25%). This excludes the jackpots, though.
This was something that could be done using pencil-and-paper arithmetic, as opposed to a lengthy computer simulation.
How?
The key insight is that a run of play on the Futurity can be broken into segments which begin when the pointer is at zero, and which end when the machine pays out a win. The percentage for one such segment can be calculated, because in each play we know the probability that it will pay out or advance to the next play, and we also know the average amount it will pay out.
So we now have a well-defined calculation of probability.

The final calculation can be worked out on paper but surely the results needed to do the calc would need thousands of samples (that's what probabilities within random groups are all about). You can't come up with a probability by running through the cycle once, which is why you need a computer to run the full set of tests first, which I believe is the reason the article in Loose Change said the designers couldn't have done the calc, it would have taken them too long!

It would take too long to work out the probabilities by experiment; that's why you do it arithmetically.

If a coin has two sides, you don't need to toss it to work out the odds. Even with many combinations of outcomes, the principle remains the same. It just takes more calculations.

Also: I've made an update to the page showing how the percentages change if the jackpot is assumed to be 100 coins. Since the change to the percentage is linear, that can be used to calculate the percentage for any value of the jackpot. Since my method of calculation appears to be novel, I felt it unreasonable to leave it as an exercise for the reader.

As Brigham above correctly states, as I'm calculating the actual percentage probability, sampling or simulating is not required. By calculating the percentage for a single full cycle from a play starting with zero on the dial, and ending with a payout, the percentage I obtain is the minimum long-term percentage obtainable by correct play. Of course a better percentage can be obtained by fortuitously finding a machine left after a few coins were played.

Simulating is simply another way to calculate the percentage if one hasn't analyzed the problem correctly first.

I sold a Bryan's Bullion to a maths teacher who used it to teach his students about probability. They did the arithmetic on different bets then tested them empirically over a long sequence of play. He said he was surprised how close the actual payout percentages came to the calculated odds. Of course a further long sequence of play might have diverged from the theoretical odds, and in practice the mechanism will always introduce biases which change over time as it wears.

Mathematically, the average payout of a Bryans Bullion on 1d is 68.57%. On 1p it is 66.29%.

Damn Decimalisation!!

quadibloc wrote: ↑Tue Jan 28, 2020 4:59 pmBy calculating the percentage for a single full cycle from a play starting with zero on the dial, and ending with a payout, the percentage I obtain is the minimum long-term percentage obtainable by correct play.

I'm sorry but that makes no sense. If you ran your single run through model three times you would get either more or less payouts. It is unlikely you would get the same, so you now have three separate and different "minimum long term % obtainable by correct play". Which one is correct? Your result only works for that one set of results.

Brighams comment about tossing a coin is also not quite correct, because he's talking about odds and probabilities as the same thing, which they are not. Even with only two outcomes, there is a slight difference (the probability is 50%, the odds are 1:1, which is not 50%). As the number of options increases, the difference expands. The following example is a common one from text books but is clearer than most:

A probability is defined as the number of occurrences of a certain event expressed as a proportion of all events that could occur. In our black bag there are three blue balls, but there are ten balls in total, so the probability that you pull out a blue ball is three divided by ten which is 30% or 0.3.

Odds is defined as the number of occurrences of a certain event expressed as a proportion of the number of non-occurrences of that event. In our black bag there are three blue balls, but there are seven balls which are not blue, so the odds for drawing a blue ball are 3:7. Odds are often expressed as odds for, which in this case would be three divided by seven, which is about 43% or 0.43, or odds against, which would be seven divided by three, which is 233% or 2.33

coppinpr wrote: ↑Wed Jan 29, 2020 12:58 pm I'm sorry but that makes no sense. If you ran your single run through model three times you would get either more or less payouts.

But quadibloc isn't running it through a model. As he said, "I'm calculating the actual percentage probability. Sampling or simulating is not required".

coppinpr wrote: ↑Wed Jan 29, 2020 12:58 pm Brighams comment about tossing a coin is also not quite correct, because he's talking about odds and probabilities as the same thing, which they are not.

We had this discussion before with regard to the gambler's fallacy. As was explained, odds and probabilities as a percentage are just two ways of expressing the likelihood of something happening in different contexts.

coppinpr wrote: ↑Wed Jan 29, 2020 12:58 pmEven with only two outcomes, there is a slight difference (the probability is 50%, the odds are 1:1, which is not 50%).

Odds of 1:1 is exactly a 50% probability.

coppinpr wrote: ↑Wed Jan 29, 2020 12:58 pmAs the number of options increases, the difference expands.

It doesn't, because the only difference is the method of expressing the chance of something happening. The chance of it happening is the same! Your textbook example is fine until the bit you added at the end:

coppinpr wrote: ↑Wed Jan 29, 2020 12:58 pmOdds are often expressed as odds for, which in this case would be three divided by seven, which is about 43% or 0.43, or odds against, which would be seven divided by three, which is 233% or 2.33

That's incorrect. To convert odds of winning into probability you divide the first number by the sum of the first and second number.
Odds for of 3:7 is 3/10 = 0.3. Multiply by 100 to express it as a percentage, we get a 30% chance of winning.
Odds against of 7:3 is another way of expressing the same thing (a 70% chance of losing).
Likewise, for a two sided coin, the odds of 1:1 convert to 1/2 - that is 0.5, or 50%.

Odds calculator