Mills Futurity - A cheating Mills Bandit?
- coppinpr
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Re: Mills Futurity - A cheating Mills Bandit?
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.
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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.
- coppinpr
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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.
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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.
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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%.
- coppinpr
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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
- coppinpr
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Re: Mills Futurity - A cheating Mills Bandit?
the example I gave was not mine, its from a text book.so, all of that must be wrong...fair enough
you say text book quote was correct except the bit i added at the end...I did NOT add anything ,I cut and pasted the complete thing,as was,no changes.
from a different on line text book (note the final line) note, I have NOT added anything
"Probability is defined as the fraction of desired outcomes in the context of every possible outcome with a value between 0 and 1, where 0 would be an impossible event and 1 would represent an inevitable event. Probabilities are usually given as percentages. [ie. 50% probability that a coin will land on HEADS.] Odds can have any value from zero to infinity and they represent a ratio of desired outcomes versus the field. Odds are a ratio, and can be given in two different ways: ‘odds in favor’ and ‘odds against’. ‘Odds in favor’ are odds describing the if an event will occur, while ‘odds against’ will describe if an event will not occur. If you are familiar with gambling, ‘odds against’ are what Vegas gives as odds. More on that later. For the coin flip odds in favor of a HEADS outcome is 1:1, not 50%."
you say text book quote was correct except the bit i added at the end...I did NOT add anything ,I cut and pasted the complete thing,as was,no changes.
from a different on line text book (note the final line) note, I have NOT added anything
"Probability is defined as the fraction of desired outcomes in the context of every possible outcome with a value between 0 and 1, where 0 would be an impossible event and 1 would represent an inevitable event. Probabilities are usually given as percentages. [ie. 50% probability that a coin will land on HEADS.] Odds can have any value from zero to infinity and they represent a ratio of desired outcomes versus the field. Odds are a ratio, and can be given in two different ways: ‘odds in favor’ and ‘odds against’. ‘Odds in favor’ are odds describing the if an event will occur, while ‘odds against’ will describe if an event will not occur. If you are familiar with gambling, ‘odds against’ are what Vegas gives as odds. More on that later. For the coin flip odds in favor of a HEADS outcome is 1:1, not 50%."
Re: Mills Futurity - A cheating Mills Bandit?
Americans express things differently. 1:1 is just another way of saying 50%.
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Re: Mills Futurity - A cheating Mills Bandit?
I can see where the confusion came from. An expression like 3:7 is another way of saying the ratio 3/7, which emphasizes that it is a ratio comparing two things and not a fraction. If at one point they say even and odd are 1:1, meaning that even and odd have the same chance of coming up, then assuming nothing else can happen than even or odd, both of them have a 50% chance.
The later part of your quote then gave the ratio 3:7. So he assumed it was 3 chances of one thing, as against 7 chances of everything else, the same way as 1:1 was. That would give 30%. But instead it was 3 chances of one thing as against all the chances, of which there were 7. So this time the ratio wasn't to the other possibilities, but to all the possibilities. Giving 3/7 or 42.8%.
The textbook you quoted wasn't "wrong", since you can use ratios to express either the ratio between two different possibilities or the chance of one possibility. But using it two different ways was at least confusing; since, usually, textbooks are nicer than that, I can see why he mistakenly blamed it on you.
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Re: Mills Futurity - A cheating Mills Bandit?
Yes, I apologize for assuming you made that bit up Paul. I just didn't believe it was in a textbook. The thing is, it wasn't - it came from this online blog, and the author (a pure mathematician) is discussing something that's not very relevant to us when he says:
Swapping our black bag with balls for a ten stop reel with 3 Cherries and 7 Lemons:
Odds for a is 3:7
Odds against a is 7:3
Probability of a is 30%
Probability of a is 70%
The additional information is that we should see 43% or 0.43 as many cherries as lemons, and 233% or 2.33 as many lemons as cherries! Gamblers would probably find this unhelpful, which is why they're not the kind of odds bookies provide.
Paul, what do you think is the percentage probability of a coin landing on heads?
Out of context this sounds like nonsense. Probabilities don't exceed 100% (except in pure maths). But he's not converting odds into probability. He's saying if you divide three by seven you get some additional information about the odds.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
Swapping our black bag with balls for a ten stop reel with 3 Cherries and 7 Lemons:
Odds for a is 3:7
Odds against a is 7:3
Probability of a is 30%
Probability of a is 70%
The additional information is that we should see 43% or 0.43 as many cherries as lemons, and 233% or 2.33 as many lemons as cherries! Gamblers would probably find this unhelpful, which is why they're not the kind of odds bookies provide.
That quote is from 'Further Reading' on the odds calculator I linked to. Again, not a textbook (accredited by a department of education), just a website (like this one). They're not contradicting their earlier reference to the "50% probability that a coin will land on HEADS". I think they're trying to say that you express the odds as a ratio of 1:1, not as 50%, which is a probability.
Paul, what do you think is the percentage probability of a coin landing on heads?
Re: Mills Futurity - A cheating Mills Bandit?
Just to throw a spanner in the works
Tossing a coin on to the ground (hence why it's usually caught in the hand) is not 1:1.
There is a possibility that it could land on the edge of the coin and sit on its side.
So what's the microscopic probability of that happening.... 1 in a million, I guess... but that would change the odds.
I'm going to go hide in the cupboard now.
All the best Dicky
Tossing a coin on to the ground (hence why it's usually caught in the hand) is not 1:1.
There is a possibility that it could land on the edge of the coin and sit on its side.
So what's the microscopic probability of that happening.... 1 in a million, I guess... but that would change the odds.
I'm going to go hide in the cupboard now.
All the best Dicky
Re: Mills Futurity - A cheating Mills Bandit?
I haven’t a clue what any of them are talking about probably why I was useless at gambling......it’s all just a jumble of numbers to me, well that is what’s use to tell my maths teacher
- badpenny
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Re: Mills Futurity - A cheating Mills Bandit?
Well, you've got me there!
Which is it then ... landing on its edge or its side?
Or is it a bit like describing a mate who lived in your road as "he and I went to different schools together"?
BP
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Re: Mills Futurity - A cheating Mills Bandit?
Mathematically derived theoretical odds are never a perfect description of real world odds. It was a surprise the Bullion came so close. The head and tail relief stamped into coins guarantees asymmetry. But if the odds were exactly 1:1, the probability of heads or tails would be exactly 50%.
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Re: Mills Futurity - A cheating Mills Bandit?
Except for being a generalization, that is pretty much true. Dice aren't perfect cubes, spinning wheels of fortune aren't absolutely perfect, and so on, and so any such device would be slightly biased. Even when great efforts are made to minimize bias, as with dice used in casinos for craps, or casino roulette wheels.pennymachines wrote: ↑Sun Feb 02, 2020 2:28 pmMathematically derived theoretical odds are never a perfect description of real world odds.
But because the bias is both small and difficult to determine, that's usually forgotten, as usually the mathematical odds are the best we can have.
Why did I mention that it's a generalization? Well, although being physical, no doubt it's still not perfect, there is a technique to deal with real-world bias. It's used in the random-number generators in microprocessors that help with picking secret code keys.
If you have an electronic noise generator that is inherently badly biased - you can't really pick a voltage and say it will be equally likely to be above or below it - instead of taking one sample to produce a random bit, take two samples for equal lengths of time and compare them.
Even for traditional games of chance, turn up two cards from a deck, and bet on whether the first or the second one is higher. Faro is an example of a game where this principle is used.
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Re: Mills Futurity - A cheating Mills Bandit?
Interesting comment about dice there.
So, what would be the odds of the dice in the window of a Buckley Bones coming up the same?
BP
So, what would be the odds of the dice in the window of a Buckley Bones coming up the same?
BP
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