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.

### Re: Mills Futurity - A cheating Mills Bandit?

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.

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.

### Re: Mills Futurity - A cheating Mills Bandit?

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!

### Re: Mills Futurity - A cheating Mills Bandit?

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.

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.

### Re: Mills Futurity - A cheating Mills Bandit?

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.

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.

- pennymachines
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### Re: Mills Futurity - A cheating Mills Bandit?

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.

### Re: Mills Futurity - A cheating Mills Bandit?

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

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### Re: Mills Futurity - A cheating Mills Bandit?

Damn Decimalisation!!

### Re: Mills Futurity - A cheating Mills Bandit?

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

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### Re: Mills Futurity - A cheating Mills Bandit?

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".

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.

Odds of 1:1 is

*exactly*a 50% probability.

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:

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

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