Sometimes I'm not sure if the wizard is being ironic

https://wizardofodds.com/games/video-poker/appendix/4/

"There is another exception that cuts the other way. With2/A/J/10/7, where the ace is suited with the jack or 10 andthe other two cards are of different suits to the royal andto each other then it is better to keep the 3 to the wildroyal, despite the penalty card. Knowing this exception willsave you one cent every $392,584.27 you put through themachine."

Are you guys REALLY getting this deep? I would have trouble creating the above page utterly straight faced. I can only presume what he's trying to say is "btw, I really, really, really like math for the sake of math, so don't take this seriously, I just wanted to show that I did more math". Is this all it is, or do plays this small really matter to some of you when you are playing optimally?

(P.S. Spell checker has shown that you missed a few spaces in the above text Michael.)

Quote: MrPapagiorgio

https://wizardofodds.com/games/video-poker/appendix/4/

"There is another exception that cuts the other way. With2/A/J/10/7, where the ace is suited with the jack or 10 andthe other two cards are of different suits to the royal andto each other then it is better to keep the 3 to the wildroyal, despite the penalty card. Knowing this exception willsave you one cent every $392,584.27 you put through themachine."

Are you guys REALLY getting this deep? I would have trouble creating the above page utterly straight faced. I can only presume what he's trying to say is "btw, I really, really, really like math for the sake of math, so don't take this seriously, I just wanted to show that I did more math". Is this all it is, or do plays this small really matter to some of you when you are playing optimally?

(P.S. Spell checker has shown that you missed a few spaces in the above text Michael.)

Some people try to play as perfect as possible.It is not really the money as in the above example but just the challenge.

You mean, you'll save a WHOLE penny?

Quote: DJTeddyBear

You mean, you'll save a WHOLE penny?

per half dollars million in.

My old obscure pages like that are often in need of reformatting. I just cleaned up that one a bit.

I have to be this is exacting. If I didn't cover that exception then I assure you that there are hundreds of perfectionists out there trying to find mistakes in everything I write. I assure you that I would hear from at least one of them if my optimal strategy were not exactly optimal. I shouldn't throw stones, as I am this way with other gambling writers I respect and will not hesitate to take them to task over any error, small or large.

Quote: Wizard

there are hundreds of perfectionists out there trying to find mistakes in everything I write.

I understand. That is often a problem with real world versus mathematical precision.

There is no way that one penny would matter after a night out at a casino. There is free flowing alcohol, there are beautiful women in low cut outfits, there is music blaring, there are slot machines jangling, there are people milling about with greasy palms and sticky fingers.

Ain't no way one penny would ever be noticed. Anyone going to a casino will need to eat and drink and will encounter people who expect to be tipped even if the room, meal and drink is utterly free. Most people go to a casino with a spouse or spouse-equivalent of some sort and the spouse's gambling will consume that one-cent. Sheer fatigue as the evening wears on can consume that one cent.

Some people memorize tables about Basic Strategy but its so much easier to adjust them slightly for mnemonic purposes or for evasion techniques.

As a gambling writer you are held to very high standards by yourself and your readership. Accounting for every penny seems an unreasonable endeavor but actually its a good goal for a mathematician, its just not so great in the real world of real people in real casinos encountering fatigue, alcohol and both player and dealer errors.

Quote: Wizard

I assure you that I would hear from at least one of them if my optimal strategy were not exactly optimal.

I hope you take it as the ultimate compliment. It only means that people regard your work as definitive, and they want to correct the little quirks. It take's no effort to find mistakes in an idiot's conclusions.

It is like that computer program that found a small error in Abramowitz and Stegun (1964), or the student who finds an error in Newton's work.


Almost every mathematician that knows craps found the error in the Time magazine article in 2009: What is the probability of rolling a pair of dice 154 times continuously at a craps table... The answer is roughly 1 in 1.56 trillion...

Quote: pacomartin

Almost every mathematician that knows craps found the error in the Time magazine article in 2009: What is the probability of rolling a pair of dice 154 times continuously at a craps table... The answer is roughly 1 in 1.56 trillion...

I show the probability of throwing the dice at least 154 times is 1 in 5,590,264,889 source. This is the probability of surviving the 153rd roll, which I show is 1.79E-10.

Stochastic processing was a big part of Anti-Submarine Warfare. If you set up the transition matrix these calculations are straightforward (assuming you have software to do matrix processing).

I know you know this Wizard, so I will post for other people.


cm 4 5 6 8 9 10

12 3 4 5 5 4 3
3 27 0 0 0 0 0
4 0 26 0 0 0 0
5 0 0 25 0 0 0 * 1/36
5 0 0 0 25 0 0
4 0 0 0 0 26 0
3 0 0 0 0 0 27

Probab of going from come out roll to a come out roll with nothing else happening
= P(7/11) + P(craps) = (6+2) +(1+2+1) = 12/36

Probability that given a point on some number, that after you roll you will still have a point on that number are:
(i.e. you won't 7-out and you won't make your point)
36 - P(7 out) - P(4) = ( 36 - 6 - 3 )/36 = 27/36
36 - P(7 out) - P(5) = ( 36 - 6 - 4 )/36 = 26/36
36 - P(7 out) - P(6) = ( 36 - 6 - 5 )/36 = 25/36
36 - P(7 out) - P(8) = ( 36 - 6 - 5 )/36 = 25/36
36 - P(7 out) - P(9) = ( 36 - 6 - 4 )/36 = 26/36
36 - P(7 out) - P(T) = ( 36 - 6 - 3 )/36 = 27/36

Raise the transition matrix to the desired number of transitions, and the sum of the first row will give you probability of returning to a "come out" state (i.e. surviving).

Initially the first row adds to (12+3+4+5+5+4+3))/36 = 100% probability of surviving past the first roll

Square the matrix
244 117 152 185 185 152 117
117 738 12 15 15 12 9
152 12 692 20 20 16 12
185 15 20 650 25 20 15 * 1/36^2
185 15 20 25 650 20 15
152 12 16 20 20 692 12
117 9 12 15 15 12 738

The first row adds to (244+117+152+185+185+152+117)=1152 so odds of surviving past the second roll are 1152/36^2 = 88.889%.

Quote: FleaStiff

I understand. That is often a problem with real world versus mathematical precision. There is no way that one penny would matter after a night out at a casino. There is free flowing alcohol, there are beautiful women in low cut outfits, there is music blaring, there are slot machines jangling, there are people milling about with greasy palms and sticky fingers.

As old as Western civilization there is distinction between theory (theoria), practice (praxis), as well as productive knowledge (techne). When I studied philosophy in school they always told us Thales of Miletus (c. 624 BC – c. 546 BC) was a the father of speculative thought. He was accused of being out of touch, so he bought all the olive presses in Miletus after predicting the weather and a good harvest for a particular year. Plato, in recounting the story, said Thales' objective was not to enrich himself but to prove his fellow Milesians that philosophy could be useful.

The other famous story is about the Gordian knot which was a well known puzzle. Alexander the Great "solved" the puzzle in an instant by cutting in two with his sword.

Look at May 28, 2009, Crunching the Numbers on a Craps Record -Wall Street Columnist. Estimates discussed in the article for the probability of throwing a dice 154 times in one turn in craps are:
(A) 1 in 1.56 trillion in Time Magazine article
(B) 1 in 3.5 billion based on overnight simulations by the Wizard
(C) 1 in 5.3 billion Wizard updated number
(D) 1 in 6.5 billion Keith Crank, manager of research and graduate education at the Amer. Stat. Ass., used a Markov chain
(E) 1 in 5.6 billion Keith Crank - updated number

Clearly there is no practical value to knowing the correct answer.
(A) Is the correct answer to the wrong problem. It is the probability of not throwing a 7 in 154 roles of a dice or 1:(36/30)^154.
(B) Simulations are very difficult when you have odds this long
(C) Not sure what Mike did here as the link is now broken. Obviously the answer is pretty close so he did a good estimate.
(D) I can tell you exactly what the mistake here was. You always get a second roll as you cannot 7-out when coming out. He raised the matrix to the 154th power
(E) He found his mistake and raised it to the 153rd power (as there are 153 transitions)

Markov Chains (Markov lived from 1856 - 1922) are analysis of discrete stochastic processes. Obviously he did his work before computers were available. However, if you can set up the matrix using very simple transform rules, you simply need to let the computer do the number crunching. Stochastic systems are those systems where the state at each moment are dependent on the previous state.

The probability that you will roll again is based on three easily calculated probabilities
(A) Probability that your come out role will be followed by another "come-out" role
(B) Probability that you will hit your point
(C) Probability that you will not hit your point , but also not 7-out

Matrix algebra was invented in the 19th century to normalize similar mathematical operations that had been done for thousands of years.

Spreadsheet available for download which I kind of like since it only has two equations.