Statistically impossible
Quote: MathExtremistThe sad thing is, if that wasn't Jerry (or Rob Singer), then there's someone else out there who acts like this. I'm going to coin a new word: antinumeracy. Worse than innumeracy, antinumeracy espouses an antipathy toward mathematics and those who understand it.
Good one. But it might apply to 95% of all kids from elementary school through high school who take math :)
While we're at it, I'll add another term, though I can't recall the source:
Teflon mind: facts cannot stick to it.
Quote: WizardDone.
See? I block Jerry 2 yesterday and today he implodes. He just couldn't stand it.
Quote: MathExtremistThe sad thing is, if that wasn't Jerry (or Rob Singer), then there's someone else out there who acts like this.
I can already feel the troll vacuum. Enjoy the peace while you can.
Quote: Nareed... Bonus question: what is Pepe the diminutive of?
Jose
Quote: DocJose
Well done!
Quote: MathExtremistThe sad thing is, if that wasn't Jerry (or Rob Singer), then there's someone else out there who acts like this.
It was Jerry, no worries. Jerry is an extreme racist, he can't help himself. Look at the racist comments in his last post, its him alright. And he baited me before I blocked him, just the way he and Singer did, with comments about how 'sad' and unhappy' I am. Nobody else does that but him/them.
Quote: DocI have a Peruvian neighbor who named his dog Paco. I don't think that was intended as racist or even derogatory in any manner. More likely intended as an endearing term.
It's an old nickname for Francisco or Frank. I didn't even know that people used it as a racist slur until I read it in the urban dictionary. While everyone in the US know Pepe, the name Paco is not that common. Paco Rabanne, and Paco Serpico are the only two people I can think of. My dog I had as a child was also named Paco.
Quote: heliciousQuote: MathExtremistThe sad thing is, if that wasn't Jerry (or Rob Singer), then there's someone else out there who acts like this. I'm going to coin a new word: antinumeracy. Worse than innumeracy, antinumeracy espouses an antipathy toward mathematics and those who understand it.
Unfortunately, what you REALLY know what's sad is how you kept after singer to support his findings from the testing he did, so he went through the trouble of telling you exactly how you could accomplish that with him in person, then you attach conditions just so you wouldn't have to face him and actually do it. Just like a weasel would do.
Some consultant. You keep tooting your horn here about what a smart math person you are, yo've got half the forum bamboozled since they wouldn't know one end of the sliderule from the other, but when you are challenged by someone with more practical mathematics application and education than you have you react by scampering out the back door like a scared puppy dog. Then your final act of cowardice is trying to belittle someone who made you feel the way you really are, and the obligatory soliciting of strength from other members for your feeble cause so you can feel good about being chicken.
JL is back- nuclear option as pre-emptive strike-
Quote: thecesspit1:42,500 (ish) is the chance to end up with a Royal after the draw playing best strategy.
That means that 3 or more royals in 8000 hands is about 1 in 900, right?
Seems pretty high. Hardly "impossible". Or did I mess it up?
Quote: heliciousUnfortunately, what you REALLY know what's sad is how you kept after singer to support his findings from the testing he did
[snip]
You just can't let it go, can you Jerry? It's really bothering you that Singer ran away from this forum as he was making his "challenge". He wasn't banned (yet), he just left. Singer never did any VP testing and doesn't have any data. I don't need to fly to Arizona to know that. The difference between you and I, Jerry, is that you're willing to take the man at his word and I'm not. I prefer to see some shred of evidence of his grand video poker conspiracy. Except nobody's seen any -- no proof of the unfair VP machines, no evidence of this flip-over theory, no support for any secret regulations, nothing. Nobody has seen any support from Singer on any of those accusations. Not even you, Jerry.
Here, Jerry, I'll give you an apple from the tree of knowledge. Whether you eat it is up to you:
Rob Singer has claimed that all VP games in Nevada operate on a safety net. He claims that their payback percentage is programmatically guaranteed to be within a range of results, and that the hands are not dealt randomly as everyone else thinks they are. Specifically, he says that VP results are intentionally programmed to be within 75% and 100% (or whatever the game's optimal payback is), rather than just programmed to converge on the payback via randomness, the way slot games and all other casino games work. Here's how you disprove his theory:
1) Buy in to a VP machine for 5000 credits. $1 or $0.25, doesn't matter. Just make sure you have 1000 plays at 5 coins each.
2) Press Bet 5.
3) Regardless of what the hand is, hold all 5 cards. Then press draw.
4) Repeat steps 2-3 for exactly 1000 hands. You must count them. Don't keep playing until you've done step 5.
5) Write down the amount on the credit meter.
6) Then do whatever you want -- cash out, keep playing, whatever.
In step 5, if the number you wrote down is not between 3750 and 5000, then Singer was wrong about the "safety net" programming.
But be careful. Don't do this test unless you're open to the idea that Singer may be wrong about his VP theories. This test will prove it.
By the way, back in January, I offered to meet Singer in a public setting in Las Vegas to disprove his safety-net theory. He refused.
At first, I thought this was a mistake / typo / whatever.Quote: MathExtremist3) Regardless of what the hand is, hold all 5 cards. Then press draw.
The reality is, it's genious.
If the machine really does have a safety net, then after losing a bunch of money by this 'bad' play, the machine will have to start dealing made hands and not rely on the draw.
Very cool.
Quote: MathExtremist3) Regardless of what the hand is, hold all 5 cards. Then press draw.
When this issue came up a few months ago, I suggested that you do even worse -- throw away dealt winners, holding only cards that would be difficult to match with winners. Eventually the safety net would need to get creative and deal winning hands that could be re-filled as winning hands on the draw no matter what you threw away.
Maybe that would overload and tire out the RNG. ;-)
6c2s2c3c9s
js4s5d6h8s
as7s7d3h8c
9h9d10d8c5d
The odds of getting those 4 hands, in that order, is about 10^24 times harder than the 3 royals. Woo hoo, I just achieved the impossible. On top of that their main accusation is that the player's pattern was indicative of bot play (high speed and consistency, no pause even after hitting the royal), in fact they even offered to pay the player to fly over and prove he could play at that speed/accuracy. Where does the statistics defying rng hack come into play?
Quote: weaselmanThat means that 3 or more royals in 8000 hands is about 1 in 900, right?
Seems pretty high. Hardly "impossible". Or did I mess it up?
Actually 40,000:1 is a better estimate.
35,939 "976-9-6" Jacks or Better
38,078 "940-9-6" Jacks or Better
39,804 "90-9-6" Jacks or Better
40,169 "9/5" Jacks or Better
40,395 "26-8-6" Jacks or Better
40,395 "8/6" Jacks or Better
40,391 "9-6" Jacks or Better
Quote: wilheimExpert testimony
4) The odds of a player hitting a single royal flush is roughly 1 in 40,000. The odds of a dealt royal flush (the player stated he received a dealt royal flush) is 1 in 649,740. The odds of hitting 3 royal flushes in 8762 hands of poker is statistically impossible. In fact in all of the years I've been in gaming I've NEVER seen that happen (and I've reviewed millions of hands of poker).
The probability of getting 3 out of 8762 is closer to 1 in 700. It is hardly impossible.
Quote: JL2No regrets.
Why would anyone have any reason to believe ANYONE lies about hitting a winning hand, especially if they put up a photo of it/them? When I look back at the Jerry Logan VERY BIG (at least to me) winners posted here, half or more of the comments center around disbelief & fraud, even though there's pictures! Is that what makes you think Mr. singer's a fraud....because a student of his posts pictures? Or maybe it's because he put up a challenge that he can prove things, and nobody had the courage to accept? I'm sure that gets the goat of people who just don't want to believe him.
Any vp jackpot win is because of luck. People who get lots of royals are very lucky people.
I do not subscribe to the line of thinking that Video Poker Royal Flushes are due to nothing more than luck. Video Poker Royals are the result of SKILL PLAY if the player has been taught, and, if the player APPLIES what he has been taught, to playing the game OPTIMALLY. I am CERTAIN that over the many, many years that I have been playing Deuces Wild, that over the many hundreds of thousands, if not millions, of Deuces Wild Video Poker hands that I have played that I have, in all probability, thrown away a Royal Flush, or Five Royal Flushes, by opting to keep the "One Deuce", Dirty Royal Flush, and, NOT ATTEMPTING TO DRAW TO A ROYAL FLUSH. This is a REASON, not the REASON, but A REASON, that I have been an overall LONG TERM NET WINNER when playing Video Poker and other advantage games. I actually go to Casinos EXPECTING TO WIN!!!
(Thank you Bob Dancer for teaching Video Poker strategy. Many thanks go to the Fiesta Hotel and Casino for hosting his classes. Also many more thanks are due to Charles Lund for publishing his book, "Robbing the One-Armed Bandits". And most importantly many thanks are also deserved by my Best Friend for enlightening me that WINNING is a PROBABILITY in Las Vegas, and even the Indian Casinos, if you know HOW and WHEN to play a game and WHAT game to play.)
On the other hand I have another friend who has hit a Royal Flush due to "Dumb Luck". He was dealt a "One Deuce", Dirty Royal Flush and then threw away the Deuce and, BY DUMB LUCK, drew to a Royal Flush. That was NOT the most "brilliant" move. In fact I cannot teach him as a result of his luck. The extra $3875 that he collected has diminished any chance that I can teach him that the actual chance that he drew to the Royal Flush was 1 in 47. Now $125 divides into $4000 is 32 times. So if you collect $125, 32 different times, then you collect the same $4000 as you would collect if you hit the Royal Flush once and miss it 46 other times.
Of course you can draw another Deuce 3 out of 47 times. You can draw to a natural straight flush 1 out of 47 times. You can draw to a flush 8 out of 47 times. You can draw to a straight 3 out of 47 times. However if you add up all of the wins by the draws then you can see why, MOST OF THE TIME, you do NOT throw away that Deuce. (The one exception to this rule is when playing "Reversible Royals" when a possible Royal Flush is set up in a way that the Royal Flush will be sequential.)
However convincing my friend that you never throw away a Deuce is a lost cause because he got the payout that he received. He has lost hundreds of thousands of dollars, if not over a million dollars, in the Indian Casinos and Las Vegas, over the past twenty years that I have known him. He refuses to play the optimal strategy on Blackjack. He does not believe in Statistical Mathematics. As a result he has NOTHING to show for his squandering of many Dollars in Casinos.
The SAD TRUTH IS that if you are not applying Statistical Mathematics to your gambling then you are being UNREASONABLE. In the opening paragraph I made a point that A REASON, NOT THE REASON, for winning is that I hold One Deuce, Dirty Royal Flushes. Well THE REASON that I am a LONG TERM WINNER is that I religiously apply Statistical Mathematics to my gambles. The Study of Mathematics "engrains one's brain" in a methodology of REASON through its rigorous study.
Logic and reason are the result of an intensive study of this enlightening language. (Yes, Mathematics is a language. It is the underlying language of the known Universe.) If one approaches anything without Mathematics at the core of the approach then one is not exercising logic and reason as the base for one's approach. If one is not exercising logic and reason; and; if one is exercising decision-making upon emotion, gut feeling, and/or other contrivances, then one is not reasonable, or, unreasonable, by definition.
Unreasonable people are the base for building Las Vegas, Nevada. Most people did not understand, or care, that when the Strip Casinos lowered their payout for Blackjack from 3:2 to 6:5 that the payouts were truly lowered. I continually have a good laugh when I see the Blackjack Tables on the Strip and Downtown filled with tourists, Casinos paying out these poor odds, and the tourists are not realizing, UNAWARE, that they are being taken advantage of, even more so!!! The Casinos make certain that their drinks are filled so that the tourists lose any sense of cognitive awareness. Yet better odds are located just a few miles away from the bustling Strip Casinos.
Well the local gamblers know better than that, so the local casinos are the destination where they go to fritter away their dollars, where the payout for a Blackjack is still 3:2. While the locals may know where the better odds exist, most of the locals, just like their tourist counterparts, do not have the capacity to understand Mathematics. So, while it may take the Local Casinos a little longer, the end results are the same...BECAUSE MOST PEOPLE DO NOT APPLY MATHEMATICS TO THEIR GAMBLING!!!
(Actually the 6:5 payout is a result that Casino Operators were realizing that since the tourist does not stay in Las Vegas, unlike a local resident, they needed to develop a way of fleecing the tourist faster than they would a local gambler. The result was the 6:5 Payout for a Blackjack. Isn't greed nice?)
However neither the tourist gambler nor the local gambler has any business gambling if they do not apply Mathematics to their gambling. It is truly sad that most gamblers haven't a real chance of winning at all. Mathematics is not valued by most of them.
Most in our society do not value Mathematics. Socially, throughout Elementary School, throughout High School, throughout post-secondary education, and, throughout life one will be ostracized as a misfit, an egghead, or a nerd if one excels in this study. Government officials and politicians will claim that there is a need for Engineers, Scientists and Mathematicians so that our country will be competitive in the 21st Century. Yet when they legislate Social Policy they make certain NOT to fund programs where these specialists would find employment. NASA and other agencies that deal with our scientific advancement were cut dramatically during the Bush Administration. The Obama Administration has followed suit with the cancellation of Project Constellation, Man's Return to the Moon and on to Mars. Furthermore they do not require a rigorous study and accomplishment throughout one's education.
Thus the pigeons line up for the foxes. The Casino Operators at one end of the spectrum are awaiting the blood fest from the uninformed gambler. I sit on the other end of the spectrum, an unemployed mathematician, waiting for the pigeons to lose their bankroll, progress to ruin, so I can feast on the scraps they left me on an optimal advantage machine. Payback is a *****!!! So I’ll continue to win because most gamblers are unreasonable.
Tall Tom
I Cor 13
Quote: MathExtremistIn fact, it's even "possible" for something to occur that has a statistical probability of zero. For example, you and I both pick a random real number between 0 and 1. The calculated probability that our numbers match is zero. But they can still match, so it's "possible".
Yes. I will agree that it is possible for something to occur which has a statistical probability of zero. That is the definition of the word, miracle.
For instance the existence of life has the statistical probability of zero. Quantum Physics has demonstrated "impossible" events such as information traveling faster than light speed.
Actually you are NOT correct when you state that is there is a probability of zero that the picks of the numbers match when two people pick a number between zero and one. I understand that an infinite number of possibilities exists between those two extremes. However what you fail to understand is that there is the limit of the Human Brain to manufacture those infinite possibilities. So since there are a finite number of possibilites due to the limits of the Human Brain then there is an actual probability that there will be a match of the two numbers thought of by two different humans.
In fact a magician's trick is to have two people pick a number between 1 and ten. Although it "should be" that each integer has a one in ten chance of being chosen, the most likely pick is 7. Try this out on your friends if you do not believe me. This is an example at how limited the Human Brain is, a "Stupid Human Trick".
One cannot ignore limits and the Calculus when one is contemplating the Mathematics of Probability. In fact the study of probability works upon the extremes of the set of Real Numbers with zero meaning an impossibility and one meaning a certainty. However it has been demonstrated that the Universe is not governed by the set of Real Numbers.
The Universe is governed by the set of Complex Numbers. Complex Numbers have a Real Component, consisting of a Real Number, and an Imaginary Component consisting of an Imaginary Number. (An Imaginary Number is a scalar multiple of the square root of negative one.) Since the Universe is governed by the set of Complex Numbers then we need to expand our Mathematics of Probability to that which is inclusive of the Complex Numbers. As the Complex Numbers allowed us to develop Mathematics which is much more descriptive of the Physical Universe then the Complex Mathematics of Probability might also be a place where Mathematicians find an expansion of ideas that will be able to explain even more REASON behind the miracle of creation. (By the numbers alone, you and I should not even exist, yet here we are.)
By the way...I did some Deuces Wild Video Poker counts at two Indian Casinos in the San Diego area last December. I lost over $4000 playing these games. I was taught by Bob Dancer so I played optimal stategy...adapting it for long term loser. (I have the money to blow.) So I actually made my counts really simple. I counted three-of-a-kind deals to four-of-a-kind draws, and four-of-a-kind deals to five of a kind draws. I kept running counts in my head.
My average draw to a four-of-a-kind was about one in thirty and my five-of-a-kind draw was about one in nineteen on the average. I know that those numbers should have been one in twenty three and a half and one in eleven and three quarters, respectively. I played about 89,000 hands, no Royals, one Set of Deuces.. I have played Video Poker since 1998 and I believe that IGT "Game King" in the San Diego region are not fair. I have not had a winning session in the San Diego Casinos for over Thrity Consecutive Sessions. The loss is generally about $100. So last December I decided to do the serious counts to see if something is up. Well I am now convinced that something is up. The Glass Pane on the Game Kings is different from those in Las Vegas, NV. They do not advertise that the game uses one 52 card deck.
I neither know who this Singer guy is, nor, do I care. The guy who was banned deserved it. But after reading the entire thread I felt that you guys need to know.
Tall Tom
I Cor 13
Quote: JL2I have no idea, and what the statistics of it are means nothing to me and most players. What I did was jump up and go to the bank both times. That's the only place numbers matter in LV. Why some have to dissect every little world event down into numbers so they can sound impressive talking about it, is a mystery to me.
Do you really believe that I do this to impress others? What a laugh!!! It means nothing to you and most players. Well I understand. Please keep on playing. You might just want to cut out the middle man, the casino, and write me the check.
Tall Tom
I Cor 13
Quote: weaselmanNo. Infinitesimal means "infinitely small". 1/1,000,000,000,000 is not infinitesimal just like 1,000,000,000,000 is not infinite.
Infinite and infinitesimal amounts cannot be expressed as real numbers.
Yes. This is called RRS syndrome, which stands for "Redundant Repetition Syndrome syndrome"
I am sorry Weaselman. Infinite and infinitesimal amounts can be expressed as Real Numbers. My Freshman and Sophmore Calculus classes only dealt with the set of Real Numbers. (The Complex Numbers were saved for Advanced Calculus.)
First here is an example of two finite Real Numbers, expressed as a Cartesian Pair, just for you: (x , y) where x is such that x is an element of the set of Real Numbers and y is such that y is an element of the set of Real Numbers. These two numbers have now been defined as Real Numbers.
So here is an example of two Infinitesimals, expressed as a Real Numbers, just for you: The Real Number dy and the Real Number dx
Here is the definition of the slope of a tangent line at any point, (x , y), on a continuous function where, y = f(x), as a Real Number which is expressed in infinitesimals: dy/dx
Understand that dy/dx = limit { f(x + h) - f(x)] / h } (letting h approach 0)
Note that if you just plug h = 0 into the equation that it becomes undefined because the rules of Algebra state that as by definition. (This rule was for good reason as the entire numbering system unravels when you divide by zero.)
However, what is so powerful about the Calculus, is that the concept of the limit was introduced as a methodology to allow one to divide an infinitesimal, which, is so close to zero that it might as well be zero for all practicality. By allowing us to do just that, what was once previously undefined gained a definition. Defining something previously undefined is the essence of creativity. Can you see just how potent this tool is? It is most amazing and wonderful...and simple...and elegant too!!!
Maybe you cannot express an infinitesimal as a Real Number but I learned how to do this trick some 32 years ago in my Freshman Calculus course. It is not too hard once you get the idea that x and y are Real Numbers and how functions...function.
Tall Tom
I Cor 13
Here is how I learned infinitesimals:Quote: TallTomMaybe you cannot express an infinitesimal as a Real Number but I learned how to do this trick some 32 years ago in my Freshman Calculus course. It is not too hard once you get the idea that x and y are Real Numbers and how functions...function.
The truth of infinitesimals
Look down into the section on "Logical Properties." There you will read about a theorem of Maltsev in 1936 in which he proved that the real numbers can be embedded in an extension field that contains objects that behave like dx and dy. It is that field extension in which calculus is really done.
Infinitesimals are neither "infinitely small" nor "so close to zero that it might as well be zero for practicality." Like many aspects of mathematics, the truth is hard to understand ... beware of slipshod mechanics and manipulation without rigor. Essentially, mathematicians had to make sense of things like dx and dy, but they did so rigorously and axiomatically.
--Ms. D.
Quote: TallTom
I am sorry Weaselman. Infinite and infinitesimal amounts can be expressed as Real Numbers.
No, they can't.
Quote:My Freshman and Sophmore Calculus classes only dealt with the set of Real Numbers. (The Complex Numbers were saved for Advanced Calculus.)
Good for you.
Quote:First here is an example of two finite Real Numbers, expressed as a Cartesian Pair, just for you: (x , y) where x is such that x is an element of the set of Real Numbers and y is such that y is an element of the set of Real Numbers. These two numbers have now been defined as Real Numbers.
You defined a real number as "an element of a set of real numbers"? Cool! :)
Quote:So here is an example of two Infinitesimals, expressed as a Real Numbers, just for you: The Real Number dy and the Real Number dx
Differentials are not real numbers.
Quote:Here is the definition of the slope of a tangent line at any point, (x , y), on a continuous function where, y = f(x), as a Real Number which is expressed in infinitesimals: dy/dx
Yes, dy/dx is (or, more precisely, may be), but dy and dx are not. Even more precisely, it is actually dy/dx + O(dy,dx), that is a real number, not dy/dx itself (you throw away the infinitesimal additives when you deal with derivatives).
Quote:Maybe you cannot express an infinitesimal as a Real Number but I learned how to do this trick some 32 years ago in my Freshman Calculus course.
Well, it's good for you that learned some tricks 32 years ago, and still remember them, but. maybe, you should have listened more closely to your teacher back then, or maybe the teacher was not very good, but you got a few wrong ideas from that course.
BTW, your idea of a definition of a miracle from one of your earlier post is also wrong. It is actually possible for zero-probability events to occur, and it has nothing to do with quantum mechanics (the "impossible events" you mentioned from there actually have non-zero probability).
Quote: gofaster87Tom, I find it hard to believe you don't know who Singer if you happen to know who Dancer is. Also, it is Dancer that says a royal is mostly luck and not skill. Funny a pro with a math background would say that.
I've known who Dancer was for years since I read about video poker. I had no idea about Singer until I came here. Now I know way more than I care to know.
Quote: thecesspitChance of 3 (or more) Royals in 9,000 hands :: 0.14%
Chance of a natural dealt Royal on one hand :: 0.000154%
Chance of hitting 3 from 10 draws on multiplay :: 0.0972%
The 3 from 10 is the most unlikely event though to see... you'll be in that position very, very rarely, whereas every time you play VP you'll be in a position to get a natural Royal. And people who play a lot of VP are going to have more than one cycle of 3 Royals (and it's even easier to happen if you only start counting from a Royal and see if you get two more in the next 8,999 hands).
I once either got option number 2, or I got 15 royals, depending how you look at it.
I was playing 15 hands out of 50 on a multiplay machine, and was dealt a natural royal.
But EasyStreet's accusations go beyond that: they claim his 3 royals in just 9000 hands were cheating! To complain that their RNG is vulnerable to tampering is a different accusation entirely. When a casino announces that their RNGs aren't working properly, they basically are telling the world that their security team is incompetent, and they can not be trusted with your money.
Quote: pacomartinIt's an old nickname for Francisco or Frank. I didn't even know that people used it as a racist slur until I read it in the urban dictionary. While everyone in the US know Pepe, the name Paco is not that common. Paco Rabanne, and Paco Serpico are the only two people I can think of. My dog I had as a child was also named Paco.
Does anyone remember Paco Stanley? When my daughter was learning Spanish, her teacher told the class to watch a Spanish television show. She used to watch Andale! starring Paco Stanley and Dos Mujeres Un Camino starring Erik Estrada who I believe had to learn Spanish himself to be on the show. She also listened to the music of Selina. Unfortunately, Erik is the sole survivor of that group. The other two were murdered.
Quote: weaselmanNo, they can't.
Good for you.
You defined a real number as "an element of a set of real numbers"? Cool! :)
Differentials are not real numbers.
Yes, dy/dx is (or, more precisely, may be), but dy and dx are not. Even more precisely, it is actually dy/dx + O(dy,dx), that is a real number, not dy/dx itself (you throw away the infinitesimal additives when you deal with derivatives).
Well, it's good for you that learned some tricks 32 years ago, and still remember them, but. maybe, you should have listened more closely to your teacher back then, or maybe the teacher was not very good, but you got a few wrong ideas from that course.
BTW, your idea of a definition of a miracle from one of your earlier post is also wrong. It is actually possible for zero-probability events to occur, and it has nothing to do with quantum mechanics (the "impossible events" you mentioned from there actually have non-zero probability).
Calculus and Analytic Geometry, by Earl Swokowski, 2nd Edition, states what a Limit is very clearly. This was the book that I learned Calculus from. I never sold it back. I refer to it many times. I read it, over and over again, for enjoyment. I know…I am strange.
By DEFINITION 2.6:
Let f be a function that is DEFINED on an open interval containing a, except possibly at a itself, and let L be a REAL NUMBER. The statement
limit(x approaches a) f(x) = L
means that for every e > 0 there exists a d > 0 such that
if 0 < |x – a| < d there exists a d > 0 such that
if 0 < |x – a| < d then |f(x) – L| < e
The previous definition is also known as the FUNDAMENTAL THEROM OF CALCULUS. This was the first Definition that I had to memorize and repeat it, verbatim, for a test. I have read it many times over. I have a very clear understanding of what it means and the implications behind it. In over thirty years I have not forgotten it as it is the FUNDAMENTAL for all Calculus. Without this definition we do not have a Calculus. I have taught it many, many times over the past thirty years.
Now let’s look at the equation that defines the derivative:
dy/dx =Limit(x approaches h)[f(x +h) – f(x)]/h
I can allow dy/dx = L , first, if dy/dx exists, and, second, if the Limit(x approaches h) [f(x +h) – f(x)]/h exists. If the equality in Definition 2.6 is true then equalities are mathematically reversible. (If a = b then b = a.) So, through a converse application of the Fundamental Therom of Calculus…By DEFINITION 2.6 if L is a Real Number then dy/dx must also be a Real Number.
Can I demonstrate that dy and dx are Real Numbers?
Well if dy/dx =Limit(x approaches h)[f(x +h) – f(x)]/h then
dy = Limit(x approaches h)[f(x +h) – f(x)]
and
dx = Limit(x approaches h) h
Again if dy/dx exists then I can infer that dy/dx is not indeterminate.
So I can set dy = L(1) and dx = L(2). Now really it does not matter what L(1) is and what L(2) is other that L(1) and L(2) are both Real Numbers by Definition 2.6 if the Limit(x approaches h)[f(x +h) – f(x)]/h exists.
So, once again I can define infinitesimals as members of the set of Real Numbers. They must be Real Numbers by definition.
Quote: TallTomIn over thirty years I have not forgotten it as it is the FUNDAMENTAL for all Calculus. Without this definition we do not have a Calculus. I have taught it many, many times over the past thirty years.
Well they had the calculus for 150 years before Weirstrauss developed the formal definition of a limit. People muddled through with a less rigorous definitions.
I actually learned Calculus from George Thomas's book which was the standard for about 25 years.
Quote: TallTom...
Well if dy/dx =Limit(x approaches h)[f(x +h) – f(x)]/h then
dy = Limit(x approaches h)[f(x +h) – f(x)]
and
dx = Limit(x approaches h) h
....
With the exception of a few typos, I followed most of your post. However, I don't get these lines at all. I think you must have this part screwed up, perhaps several times.
Quote: gofaster87Tom, I find it hard to believe you don't know who Singer if you happen to know who Dancer is. Also, it is Dancer that says a royal is mostly luck and not skill. Funny a pro with a math background would say that.
Actually I do not know who Singer is. I really have not been involoved in the Video Poker World. I stumbled on to this site as a response to a request from a friend about percentages of dealt winning hands on Deuces Wild.
However, from what I remember, and I have a really good memory if I remember correctly : - ), Bob Dancer said in his class in October, 1998, at Fiesta Hotel and Casino, that, "The first two hours of play can be attributed to luck while longer term play is attributed to skill." I do not ever recall him say that if the game is played optimally then Royal Flushes were due to luck.
Anyway I found that some slot games were far better payers in certain conditions than Video Poker was. By reading information on the screen, you could determine that playing those particular games would be to your great advantage. Charles Lund wrote a book, "Robbing the One Armed Bandits". He was a mathematician out of University of Maryland and his statistical work allowed me to learn what games to look for and under what conditions playing was advantageous. I netted a "few dollars" playing these games. : - )
Do I want to know who Singer is ? From what I gather he is somewhat controversial. I have done some running counts in my head and I have found that the Indian Casino VP games are not quite right. If that is what he is claiming then I believe that he may have a point that might be considered. Game King by IGT ate me alive. I counted dealt three of a kinds and dealt four of a kinds yielding four of a kinds and five of a kinds, respectively, on Deuces Wild at two local Indian Casinos in San Diego County. The expected numbers and the actual numbers were substantial. I lost $4000 net in December. I am a believer in optimal play. I have VP tutor and set the game to the losing schedules. I practiced before I played. My opinion, take it for what it is worth, is that the VP games are not fair. Just know that before walking in the door.
That is based on the fact that you lost, I assume,
Quote: DorothyGaleHere is how I learned infinitesimals:
The truth of infinitesimals
Look down into the section on "Logical Properties." There you will read about a theorem of Maltsev in 1936 in which he proved that the real numbers can be embedded in an extension field that contains objects that behave like dx and dy. It is that field extension in which calculus is really done.
Infinitesimals are neither "infinitely small" nor "so close to zero that it might as well be zero for practicality." Like many aspects of mathematics, the truth is hard to understand ... beware of slipshod mechanics and manipulation without rigor. Essentially, mathematicians had to make sense of things like dx and dy, but they did so rigorously and axiomatically.
--Ms. D.
Thank you. I will read about this and consider this. In Real Number Mathematics, by definition, the infinitesimal is a Real Number. If it is not then what is it, what type of number is it? I believe that Real Number mathematics is not the best understanding of Mathematics. As I previously wrote the Mathematics of Complex Numbers has enlightened Mathematicians to "truths" that Real Number Mathematics could not. The study of Mathematics is far from complete. Until I find a reason to revise my understanding then an infinitesimal is PRACTICALLY zero. That means if you are to apply Mathematics to Real World phenomena then, in practice, you can consider an infinitesimal zero.
For instance in Newtonian Physics how does one determine "Instantaneous Velocity" where zero time happens? Well v = dx/dt where dt, the infinitesimal, is zero time for "practical" purposes.
But I understand that time is quantum in nature. You either have a chronon, the quantum unit of time, or you don't. So, really, a function that deals with time is not continuous. And one cannot determine a limit on a discontinous function. So our Fundamental Therom of Calculus does not apply on the quantum level. (I like how the belief that nothing with the property of mass can travel faster than light speed might be erroneous because the Energy function is not continuous on the quantum level. It gives me hope for the possibility of Interstellar Travel.) Yes there are many problems. So, as a result, there are many opportunities.
Quote: DorothyGaleHere is how I learned infinitesimals:
The truth of infinitesimals
Look down into the section on "Logical Properties." There you will read about a theorem of Maltsev in 1936 in which he proved that the real numbers can be embedded in an extension field that contains objects that behave like dx and dy. It is that field extension in which calculus is really done.
Infinitesimals are neither "infinitely small" nor "so close to zero that it might as well be zero for practicality." Like many aspects of mathematics, the truth is hard to understand ... beware of slipshod mechanics and manipulation without rigor. Essentially, mathematicians had to make sense of things like dx and dy, but they did so rigorously and axiomatically.
--Ms. D.
Dorothy, pay not attention to the man behind the curtain. YOU ARE THE WIZARD OF KANSAS !!
