Analysis of an Antique Slot Machine
In a recent post MathExtremist said of me "By your own admission, you are relatively new to a great deal of the technologies used in analysis of casino games." That's not quite true. I have been interested in gaming math for quite some time and have done extensive work on blackjack strategies. I recomputed Thorp's dealer probabilities entirely by hand and found it accurate. In the course of doing so I discovered that Thorp's deck was complete except for the dealer's up card. Thorp used an IBM 7090 computer at MIT. The page linked to above is about eight years old.
I think MathExtremist considers me naive because I distrust simulations and prefer exact analysis. In a later post I may explain why. I gather that they are the principal tool of the crowd here.
I played a machine like the one above at a hotel in pre-Castro Cuba. I put in quarters but they paid off in a Cuban coin that was worth slightly less than a quarter. After Castro took over the hotel was raided by a mob that smashed the machines. I think that was less of a political statement against the concessionaire, Meyer Lansky, than that they wanted to get at the money.
Quote: statmanAnalysis of an Antique Slot Machine
In a recent post MathExtremist said of me "By your own admission, you are relatively new to a great deal of the technologies used in analysis of casino games." That's not quite true. I have been interested in gaming math for quite some time and have done extensive work on blackjack strategies. I recomputed Thorp's dealer probabilities entirely by hand and found it accurate. In the course of doing so I discovered that Thorp's deck was complete except for the dealer's up card. Thorp used an IBM 7090 computer at MIT. The page linked to above is about eight years old.
I think MathExtremist considers me naive because I distrust simulations and prefer exact analysis. In a later post I may explain why. I gather that they are the principal tool of the crowd here.
I played a machine like the one above at a hotel in pre-Castro Cuba. I put in quarters but they paid off in a Cuban coin that was worth slightly less than a quarter. After Castro took over the hotel was raided by a mob that smashed the machines. I think that was less of a political statement against the concessionaire, Meyer Lansky, than that they wanted to get at the money.
Quoted for preservation.
From what I understand from an earlier thread, the reason ME considers you naive is exactly the opposite - trying to substitute analysis with simulation looking for a bias in a roulette wheel (which is particularly weird - attempting to simulate a distribution that is unknown).
I like it. Considering Statman's prior erratic forum behavior, I like it a lot.Quote: weaselmanQuoted for preservation.
That said, I too was going to point out the incongruity of distrusting simulations while touting the results of the analysis of a simulated roulette wheel that has a programmed bias.
Quote: statmanThe page linked to above is about eight years old.
So is the Wizard's page about this same game, but with more detail. It is also the easiest game ever to analyze and it took me roughly 3 minutes.
Quote: statmanI played a machine like the one above at a hotel in pre-Castro Cuba. I put in quarters but they paid off in a Cuban coin that was worth slightly less than a quarter. After Castro took over the hotel was raided by a mob that smashed the machines. I think that was less of a political statement against the concessionaire, Meyer Lansky, than that they wanted to get at the money.
These old time slot machines were so straightforward. There is nothing seductive built into the payoff schedule.But with the odds of payoff of 100:1 or over only once in 421 plays, they must have been easy to walk away from.
Quote: statman
I think MathExtremist considers me naive because I distrust simulations and prefer exact analysis. In a later post I may explain why. I gather that they are the principal tool of the crowd here.
Nah, they appear to use a variety of methods for analysis, either simulation, static analysis or best fit. Depending on what tool suits the job, and the person doing the works preference. I've seen three posters use three different approaches and get the same answer.
I'm guessing your distrust of simulation is something to do with imperfect RNGs and cycling of those RNGs (based on other posts).
Quote: statmanI think MathExtremist considers me naive because I distrust simulations and prefer exact analysis. In a later post I may explain why. I gather that they are the principal tool of the crowd here.
Almost all analysis done by the Wizard does not involve a simulation. Simulations are good in a time crunch, or if you don't have enough data. But as a mathematician I prefer exact answers.
Quote: statmanI think MathExtremist considers me naive because I distrust simulations and prefer exact analysis. In a later post I may explain why. I gather that they are the principal tool of the crowd here.
I prefer using the right tool for the job, but unlike you I do not distrust simulations. Monte Carlo analysis is just another tool in the toolbox. Not all questions are amenable to efficient closed-form solutions. Also, I prefer being able to double-check my work with multiple methodologies.
When I do a new card game, I almost always write an iterative analysis as well as a MC simulator. I get worried if they don't match.
I have been looking at some PAR's, which MathExtremist put me on to, and I find them quite boggling. I'll leave those to the younger guys.
weaselman and DjTeddyBear: with respect to Murphy's Extreme Bias Roulette Wheel, I resorted to a simulation because the information required to do an exact analysis is lacking. The distribution is the multinomial distribution in 38 variables but in order to use it you have to know the theoretical probabilities of every number on the wheel. Murphy gives the number of hits on only three numbers. This make possible estimates of their probabilities, which probably are close because of the large number of trials, but tells us nothing about the probabilities of the other numbers. Trying to read from Murphy's polar chart would not be very precise. I'll write to Murphy and ask if he would be willing and able to provide the number of hits on all the numbers of the wheel. If the probabilities of some numbers are above average, those of other numbers must be below average and that would add to the value of X2. The more extreme the deviation the greater the contribution to the value of X2.
Even though the model I used might not be a very good model of Murphy's wheel, it is a model of some biased wheel that might possibly exist and the point was to find out how reliable the X2 test is in detecting bias with a reasonable number of spins. Even 1,000 spins would be beyond the data collection ability of most players. It is the test that was recommended by the Wizard when a Macau player submitted data and asked an opinion of whether or not a particular wheel was biased. My conclusion was that it was not a reliable test because using Murphy's value of X2 = 55 or higher as indicating bias, it could call a wheel biased even when the most frequently occurring number was not a biased one. That is a false indication because although the wheel really is biased, the frequency data would not give the player the right number to play and his fortune could go down the drain. I think that is a useful discovery.
Quote: statmanThanks, guys, especially CrystalMath, for not bashing me. Does that mean I got it right? I delete posts only if they draw heavy fire, which to me means that they should not stay up to draw even more fire and distract the forum members from more productive pursuits. Could you provide a link to the Wizard's page about this game?
If you can't stand by what you wrote, you shouldn't post it in the first place.
Writing something, having it critiqued and removing it again, rather than defending it, or updating later if there's a mistake is poor etiquette and makes it look like you have something to hide.
Quote: statmanCould you provide a link to the Wizard's page about this game?
It is here
I actually found it by following the links on your website.
Quote: statmanThanks, guys, especially CrystalMath, for not bashing me. Does that mean I got it right? I delete posts only if they draw heavy fire, which to me means that they should not stay up to draw even more fire and distract the forum members from more productive pursuits. Could you provide a link to the Wizard's page about this game?
I have been looking at some PAR's, which MathExtremist put me on to, and I find them quite boggling. I'll leave those to the younger guys.
weaselman and DjTeddyBear: with respect to Murphy's Extreme Bias Roulette Wheel, I resorted to a simulation because the information required to do an exact analysis is lacking. The distribution is the multinomial distribution in 38 variables but in order to use it you have to know the theoretical probabilities of every number on the wheel. Murphy gives the number of hits on only three numbers. This make possible estimates of their probabilities, which probably are close because of the large number of trials, but tells us nothing about the probabilities of the other numbers. Trying to read from Murphy's polar chart would not be very precise. I'll write to Murphy and ask if he would be willing and able to provide the number of hits on all the numbers of the wheel. If the probabilities of some numbers are above average, those of other numbers must be below average and that would add to the value of X2. The more extreme the deviation the greater the contribution to the value of X2.
Even though the model I used might not be a very good model of Murphy's wheel, it is a model of some biased wheel that might possibly exist and the point was to find out how reliable the X2 test is in detecting bias with a reasonable number of spins. Even 1,000 spins would be beyond the data collection ability of most players. It is the test that was recommended by the Wizard when a Macau player submitted data and asked an opinion of whether or not a particular wheel was biased. My conclusion was that it was not a reliable test because using Murphy's value of X2 = 55 or higher as indicating bias, it could call a wheel biased even when the most frequently occurring number was not a biased one. That is a false indication because although the wheel really is biased, the frequency data would not give the player the right number to play and his fortune could go down the drain. I think that is a useful discovery.
Quoted for preservation.
If you don't know the distribution, you can't simulate it. If you do know the distribution, you don't need to simulate it, because you can find an analytical solution.
Quote:weaselman: If you don't know the distribution, you can't simulate it. If you do know the distribution, you don't need to simulate it, because you can find an analytical solution.
The distribution is well known, it's the multinomial distribution in 38 variables, but for this many variables it is not resorted to because it is extremely time-consuming to calculate. Calculating the coefficients of just one term involves dividing a large factorial by the product of 38 smaller factorials. Even calculating one factorial can take quite a few machine cycles.
It would be possible to evaluate the trueness of a roulette wheel by calculating the multinomial term based on the record of the wheel and comparing it to the corresponding term of a true wheel, but nobody does this because the X2 test requires much less calculation, even though it is approximate. It is not considered especially accurate for a small number of trials and a small number of variables, but for those cases it is practical to use a multinomial distribution.
It's gratifying to see you on your best behavior.
Upon re-reading prior posts, I decided my comment wasn't required.
