Analysis of an Antique Slot Machine

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October 18th, 2011 at 2:51:55 PM permalink
CrystalMath Member since: May 10, 2011 Threads: 3 Posts: 474	Quote: statman Could 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. I heart Crystal Math.

October 18th, 2011 at 3:29:00 PM permalink

weaselman

Member since: Jul 11, 2010
Threads: 16
Posts: 1918

Quote: statman
Thanks, 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.

"When two people always agree one of them is unnecessary"

October 20th, 2011 at 1:45:33 AM permalink

statman

Member since: Sep 25, 2011
Threads: 12
Posts: 95

CrystalMath: Thank you for the link. The Wizard may have gotten the data the same place I did: Scarne's Complete Guide to Casino Gambling. The Wizard's major work was the analysis of the machines at Vegas's major locations. I wondered how he had done it without spending a fortune playing the machines. It seems he managed to get his hands on PARs despite some opposition to his survey. Good for him!

A fool is someone whose pencil wears out before its eraser does. - Marilyn Vos Savant

October 20th, 2011 at 5:31:32 AM permalink
Doc Member since: Feb 27, 2010 Threads: 20 Posts: 2792	I'm not certain, but it is my impression that Mike actually played the machines and recorded the observed portions of the reels (everything visible) sufficiently to generate his own data for the sequences and quantities of each image on each reel. I don't know on how many machines he did this.

October 21st, 2011 at 6:04:28 AM permalink

statman

Member since: Sep 25, 2011
Threads: 12
Posts: 95

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.

A fool is someone whose pencil wears out before its eraser does. - Marilyn Vos Savant

October 21st, 2011 at 6:45:19 AM permalink
DJTeddyBear Member since: Nov 2, 2009 Threads: 105 Posts: 5691	Deleted. Upon re-reading prior posts, I decided my comment wasn't required. Superstitions are silly, childish, irrational rituals, born out of fear of the unknown. But how much does it cost to knock on wood?

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