The scams of Statistics...

What is the basis for reason? And mathematics?

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Scott Mayers
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Re: The scams of Statistics...

Post by Scott Mayers »

Obvious Leo wrote:Scott. Why do you keep referring to scenarios which are not the Monty Hall scenario under discussion? If you would confine yourself to simple yes/no answers to my questions we might be able to get somewhere but once you start qualifying those answers we are no longer talking about the same thing.
I want to do this step by step in the simplest way possible.

Do you agree that in MY experiment if I place a single coin under a single cup you have a 1/3 chance of guessing which cup I placed it under. Forget about what comes next and just answer yes or no.
I don't mind trying to appealing to your request but have already done this where you don't see the agreement. It's like if I claimed the both Joe and Cheryl are good and valid people but you think that somehow a likeness of these individuals are exclusive. So you try to prove to me that Joe is a good and valid person, something I already know. I point to the fact that you must be thinking that a belief in Joe necessarily requires an opposing exclusion of belief in Cheryl by your intent to try to prove that somehow I don't understand Joe's validity.

Then I DO try to prove that Joe is valid for your sake. But you can't relate to my method(s) of proving anything one way or the other and so keep demanding that I follow your lead. When I use all the various different approaches (different methods), I'm trying to demonstrate that I understand you but I keep getting rebuffed by you declaring that ONLY ONE way will suffice to prove to you that Joe is a valid person.

dionisos is doing the same thing when he demands that I only appeal to proving to him anything by opting for one particular route (method) to get to some goal. But the very fact that one is so restrictive to only see one unique route by default begs that no other such route exists.

Given town A to begin with, and aiming to get to town B has a road, R1, in which you and most people are familiar with. I argue that there is another Rx that exists to get to town B AND that there are also other towns that exist, like town C, in which we can go to that provide the same VALUE for seeking any town by meaning.

For instance, if your reason for going to town B is because it has a concert performance by your favorite artist, I might be noticing that this same artist is performing in another town even closer. But you deny both that such a town exists AND that no other route is possible to go regardless. So all you want me to do is to validate your preferred restriction with absolution and stop talking about these other towns or routes.

So be it. I'm not the one who is denying your right to your given goals or the routes. But you are demanding me to both AGREE with you absolutely AND shut up about any other matters because they are not important to you. But I began this thread. And while I am welcoming of your presence, you are coming to MY party and demanding that I abandon it to come to your home to party instead. I don't mind your own invite. The problem is, you are spoiling my party to divert others to go to yours instead and it acts to invalidate me as a person.

I like both Joe and Cheryl. And if you come to my party but don't like Cheryl's presence, you either have to depart without prejudice or treat me and my company with the same respect I'd grant you. Otherwise, you are appearing to offend me. You apologized and I can accept this. But if I'm complaining that you are stepping on my toes and you keep apologizing but STILL step on my toes repeatedly, how am I to interpret even your apologies as being sincere?
Obvious Leo
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Re: The scams of Statistics...

Post by Obvious Leo »

Does this mean that you decline to answer my question? It was you that introduced the topic of the Monty Hall puzzle and yet you steadfastly refuse to discuss it. What do you hope to gain from such a conversation? My apology was sincere but my frustration is genuine because you are speaking to various hypothetical scenarios which are not analogous to the one you introduced.

This is a very simple proposition of formal logic. If I offer you a guess as to which cup conceals the coin you have a 1/3 chance of being right. This means that there is a 2/3 chance that the coin is concealed beneath one of the two cups which you haven't guessed. The rest of the story is just flim-flam designed to confuse you.

Kindly point out in the simplest possible language why this conclusion is false.
Scott Mayers
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Re: The scams of Statistics...

Post by Scott Mayers »

Obvious Leo wrote:Does this mean that you decline to answer my question?
Yes, I respectfully decline. As HOST of this thread, I am revealing a door marked "exit" and am asking that you take it without prejudice.

Thank you.
mickthinks
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Re: The scams of Statistics...

Post by mickthinks »

Scott, if you are still here,

If I presented a box to you without you being able to see inside it, it is indeterminate to you.

Yes, and of course we can't use the mathematics of probability if we have no reliable information or can make no reasonable assumptions. However, when we see a 10p coin being tossed we have some reliable information and when we are offered three cups from which to chose the one covering the pea, it is usually reasonable to assume that one and only one cup conceals a pea. In such cases we can determine the odds of the events "coin shows head" and "cup conceals pea" (1/2 and 1/3 respectively). The odds are not indeterminate even while the outcome maybe.

The odds to play a game once is like buying a lotto ticket given some odds. In such a case, you may also be indeterminate even being told of the supposed odds since it could be rigged.

Of course, when there are grounds for doubting the truth of the information we have been given, the odds may be impossible to determine. The real Monty Hall game may have been rigged in ways we don't know about. But this puzzle is about a form of the Monty Hall game in which we are asked to devise the best strategy for winning the car assuming the information we have been given about the game is true. The odds are determined, just as they are in the cases of the coin and the cup with the pea, and can be calculated mathematically on that basis, and the winning strategy is always to switch to the remaining door after the location of one of the goats has been revealed.


Having trust in the fairness is not sufficient enough to determine if the odds are fair.
I don't understand this sentence, so if you think it is important perhaps you'd like to try putting it another way. I'm happy to let it go, though. :-)
Scott Mayers
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Re: The scams of Statistics...

Post by Scott Mayers »

mickthinks wrote:Scott, if you are still here,

If I presented a box to you without you being able to see inside it, it is indeterminate to you.

Yes, and of course we can't use the mathematics of probability if we have no reliable information or can make no reasonable assumptions. However, when we see a 10p coin being tossed we have some reliable information and when we are offered three cups from which to chose the one covering the pea, it is usually reasonable to assume that one and only one cup conceals a pea. In such cases we can determine the odds of the events "coin shows head" and "cup conceals pea" (1/2 and 1/3 respectively). The odds are not indeterminate even while the outcome maybe.
Of course if we saw one toss the coin it would assure us, if the host was not 'faking' it, that the pea is under our selected cup. The puzzle though prevents us from knowing when or where the host does this in his head. We'd have to at best default to assume that he does this toss for each 1/3 possibility. Then you have to also multiply these 1/3 probabilities by 1/2 = 1/6 for each of them. And then we only represent the case where the host can only reveal the car and eliminate out the cases where he does. This then makes each case 1/6 to win the car, or collectively as 1/3 total for switching. And each case already where you pick the car, to switch is already understood as 1/3 [1/6 + 1/6]. so 1/3 to Win: 1/3 to Lose or 1/2 probability for each.

Do you want an illustration?
The odds to play a game once is like buying a lotto ticket given some odds. In such a case, you may also be indeterminate even being told of the supposed odds since it could be rigged.

Of course, when there are grounds for doubting the truth of the information we have been given, the odds may be impossible to determine. The real Monty Hall game may have been rigged in ways we don't know about. But this puzzle is about a form of the Monty Hall game in which we are asked to devise the best strategy for winning the car assuming the information we have been given about the game is true. The odds are determined, just as they are in the cases of the coin and the cup with the pea, and can be calculated mathematically on that basis, and the winning strategy is always to switch to the remaining door after the location of one of the goats has been revealed.


Having trust in the fairness is not sufficient enough to determine if the odds are fair.
I don't understand this sentence, so if you think it is important perhaps you'd like to try putting it another way. I'm happy to let it go, though. :-)
Basically, nature itself is also 'unfair' regardless. That is, if it were 'fair', randomness would merely be an illusion by perspective only. That's the major reason why I don't believe we can use repeated experiments to measure this.
Obvious Leo
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Re: The scams of Statistics...

Post by Obvious Leo »

Scott Mayers wrote:That's the major reason why I don't believe we can use repeated experiments to measure this.
Is this truly a claim you wish to make? Every single time the Monty Hall experiment has actually ever been performed it has been shown that switching choices doubles the chances of winning. Are you saying that this experimental outcome is valueless because there exist a set of mathematical equations which say that switching choices has no effect on the outcome?
Scott Mayers
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Re: The scams of Statistics...

Post by Scott Mayers »

Obvious Leo wrote:
Scott Mayers wrote:That's the major reason why I don't believe we can use repeated experiments to measure this.
Is this truly a claim you wish to make? Every single time the Monty Hall experiment has actually ever been performed it has been shown that switching choices doubles the chances of winning. Are you saying that this experimental outcome is valueless because there exist a set of mathematical equations which say that switching choices has no effect on the outcome?
When I was arguing with dionisos, I showed how the way a program is designed can expect what it is seeking by its coding. I studied (and still do) computer architecture and in this each step costs time. In a program that measures only the RESULTS without respect to time, it will appear as 2/3. But in reality, nature limits certain processes, as you already know.

If you use a die to roll for the first toss where you assume a '1' or '2' as Door 1, a '3' or '4' as Door 2, and a '5' and '6' as Door 3, then you can 'toss' the die to determine which of the three doors both the host places the Car in and for the Guest to Select initially. Ignoring the placing of the Car, from the Guest perspective, he has ONE event, the die roll for the first step. But only if the door is picked initially which has the Car, the host must do another event, a coin toss (in his head or literally), to which costs time to do. So when you have the Car (unknowingly) there are 2 times (rather than 1/2) the events that the processor must use to toss that coin for the end result = 2/3. But this is the same time to determine both of the remaining choices and switch.

Thus, the standard measure to determine the 2/3 solution refers to granting the measure of probability respecting the game as a whole, while the 1/2 solution refers to granting the measure of probability of each result respecting time. Thus, if it takes 1 second to toss a die in the first selection, prior to switching involves another second to toss the coin in the case one picks the car. This means that if you repeat the game a thousand times, only considering wins per game, this would be 2/3. But if you then divide this with respect to the time needed for each game, you get the 1/2 result.

I also showed earlier how even simply pretending to switch but then ask from that imagined place what the odds are to win by 'switching back' you'll see that this too is 2/3 to win the car! So it is an illusion. Since the 1/2 relation is true, this means that it is a 1:1 relationship. Because of this, any multiple of these is possible with regards to nature just as it is possible even if less apparently probable to toss heads one million times in a row.
Obvious Leo
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Re: The scams of Statistics...

Post by Obvious Leo »

Scott Mayers wrote:Since the 1/2 relation is true,
The problem with your theory is that the 1/2 relation is not true, Scott, and the sad fact is that the most magnificent of beautiful theories can be undone by a single inconvenient fact.

How are you ever going to deal with the FACT that switching choices doubles the chances of winning the car because the FACT that this is a FACT is a FACT beyond dispute. In the nineties every high school kid in Australia was required to confirm this FACT for themselves as a homework assignment and I'd be very surprised if this wasn't the case all over the world. How do you propose to address this problem?
Scott Mayers
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Re: The scams of Statistics...

Post by Scott Mayers »

Obvious Leo wrote:
Scott Mayers wrote:Since the 1/2 relation is true,
The problem with your theory is that the 1/2 relation is not true, Scott, and the sad fact is that the most magnificent of beautiful theories can be undone by a single inconvenient fact.

How are you ever going to deal with the FACT that switching choices doubles the chances of winning the car because the FACT that this is a FACT is a FACT beyond dispute. In the nineties every high school kid in Australia was required to confirm this FACT for themselves as a homework assignment and I'd be very surprised if this wasn't the case all over the world. How do you propose to address this problem?
It's not a problem to me. There was a time too that every child in the British Empire had to respect the Queen...oh wait, we still do, ...never mind. :oops:

P.S. Think of the point I made about the odds of any person winning a lottery versus the odds of one particular person winning. You can't use popularity to discern this probability.

If you were to think of the real time it takes to play a game as winning every four cycles average through time, in the time it would take to play 3 games at 2/3 odds, you only win half the time you invest. That is, if 3 games took an hour to play, and you get paid by the wins, you'd get a probability of 2 dollars. But if you were paid a dollar for every toss + dice rolls that last in equal measures, you'd still get 2 dollars. But this is $2/(3 games) OR $2/($4 possible dollars you could make on tosses). It's about perspective only. But the maximum you could win in the 3 game odds of 2/3 only assures the dealer only has to pay out a maximum of $3. In contrast, I bet on the 1/2 and would get $4 maximum in the same investment.

What matters to you is the odds of winning the car in any one game. Thus it is reasonable to think of switching for the two thirds chance of winning. But if you value your time, imagine if each game requires 10 years to play. [Perhaps the odds required even to get on a show like "Let's Make a Deal".] You could sacrifice the time to play the game for this duration in order to assert your best odds to win as 2/3 but in reality, you would be better off getting a job that pays you an income that could earn the car plus 1/2 of the down-payment for a new future car (as savings) in the same period! Only, I'd bet on the Goats when I switched as I'd get $2 x 3 games = $6 possible maximum!!

This may be a mere technicality I present on perspective, but the point is, no one would actually ever even allow you to play such a real game at 2/3 odds to win for every $1.00 you might invest in each game. I could charge you $1.33 per game and it becomes 1/2 odds to win your dollar back. Either way, you'd still be the fool to play. The return on our provincial lottery machines for a player is 85%! This tricks people into thinking the investment is worth while. Yet, what they don't realize is that this still assures the profit is average of 15% for the house's favor over multiple plays.
mickthinks
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Re: The scams of Statistics...

Post by mickthinks »

Scott,

We'd have to at best default to assume that he does this toss for each 1/3 possibility. Then you have to also multiply these 1/3 probabilities by 1/2 = 1/6 for each of them.

We can do that and show the results like this:

Code: Select all

1st pick	Monty tosses a coin	2nd pick	probabilty
				and reveals
				
						Goat 2		-	 Car		   1/6	
  Goat 1 <
						Goat 2		-	 Car		   1/6
 			 	
 			 							 
						Goat 1		-	 Car		   1/6	
  Goat 2 <
						Goat 1		-	 Car		   1/6	
 			 
 			 
						Goat 1		-	 Goat 2	   1/6
  Car	 <	 							 
						Goat 2		-	 Goat 1	   1/6
Basically, nature itself is also 'unfair' regardless. That is, if it were 'fair', randomness would merely be an illusion by perspective only.

You've thrown a new concept into the pot here and I am not sure what you are trying to say. Do you mean that, since a random event does not always produce the most likely outcome, that makes real life unfair? If that is what you mean, I think you are confusing probability with fairness. No one claims to have calculated a way we can win every time. What the mathematics of probability gives us is a strategy for improving the odds of us winning. The important thing to notice is the difference between "winning" and "increasing the probability of winning".
Scott Mayers
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Re: The scams of Statistics...

Post by Scott Mayers »

mickthinks wrote:Scott,

We'd have to at best default to assume that he does this toss for each 1/3 possibility. Then you have to also multiply these 1/3 probabilities by 1/2 = 1/6 for each of them.

We can do that and show the results like this:

Code: Select all

1st pick	Monty tosses a coin	2nd pick	probabilty
				and reveals
				
						Goat 2		-	 Car		   1/6	
  Goat 1 <
						Goat 2		-	 Car		   1/6
 			 	
 			 							 
						Goat 1		-	 Car		   1/6	
  Goat 2 <
						Goat 1		-	 Car		   1/6	
 			 
 			 
						Goat 1		-	 Goat 2	   1/6
  Car	 <	 							 
						Goat 2		-	 Goat 1	   1/6
Basically, nature itself is also 'unfair' regardless. That is, if it were 'fair', randomness would merely be an illusion by perspective only.

You've thrown a new concept into the pot here and I am not sure what you are trying to say. Do you mean that, since a random event does not always produce the most likely outcome, that makes real life unfair? If that is what you mean, I think you are confusing probability with fairness. No one claims to have calculated a way we can win every time. What the mathematics of probability gives us is a strategy for improving the odds of us winning. The important thing to notice is the difference between "winning" and "increasing the probability of winning".
I see that you're following my route of interpretation. What I noticed from the above code as you see too is that if we have any goat on the initial selection we get the car. But it can also be understood as this too, mathematically:

Code: Select all

1st pick	Monty tosses a coin	2nd pick	probabilty
				and reveals
				
						Goat 2		-	 Car		   1/6	
  Goat 1 <
						[Car]		-	  [Goat 2]		   [1/6]
 			 	
 			 							 
						Goat 1		-	 Car		   1/6	
  Goat 2 <
						[Car]		-	 [Goat 1]		   [1/6]	
 			 
 			 
						Goat 1		-	 Goat 2	   1/6
  Car	 <	 							 
						Goat 2		-	 Goat 1	   1/6
What I bracketed above can be understood as what option is NOT actually available but counts if we have to 'fake' the toss in the host's mind for those when the Guest selects any goat. That is, the options above are the 'real' options even though the Host could not select the ones in the brackets. Note that a computer can be programmed to favor the way you demonstrated or the second one I did. In both cases, the Host already knows where the car is and learns whether the Guest selects the correct door or not. So, for a computer to do this, it takes the same amount of effort for each up to that point. However, in the first case, a computer can either 'fake' the toss unnecessarily and would thus result in the 2/3 as your demonstration applies.

In the second case (my added code demonstration), it saves the time to bother with the ones in brackets as they are not possible. Obviously, you can see that the second case would reduce the odds to 1/2.

Now by the perspective of a Guest, they cannot determine whether the Host tosses. A simple example is if the Host simply always simply defaults to favoring the left-most door by some nature in his head to favor leftness. So the host might for instance choose to always select the left-most door when he has options (when the Guest selects goats, that is). In this way, since the Guest cannot determine whether the Host actually 'flips a coin' or simply always opts to choose the left door. This too reduces to the 1/2 as my altered suggestion. Thus the real odds are indeterminate as being between 1/3 and 1/2.

This is why you can't trust allowing a computer program to do this in online statistics for games. It COULD be done such as the program tosses a coin to determine between two of these kinds of programs. So a program CAN be designed to include both code forms and but only opt for one randomly. Then, you should likely get an average of (1/3 + 1/2)/2 = (2/6 + 3/6)/2 = (5/6)/2 = 5/12 average wins in some online samples.

Can you see what I'm getting at here?

When I first learned the puzzle, I saw the 2/3. But by the time I re-investigated this, I also had learned much of logic, especially computer logic, which raised issues about timing to DO a function. This costs time. For a simple example, in computer logic, an OR gate might be recognized to be easily created by merely connecting three wires together at one point. But because a negation gate, and AND gate costs more to create in nature, they redesign the circuits to be more complex in such a way to make the timing for each gate equal in time. It makes some of the circuitry redundant but is necessary or the computer would go out of sync because each gate takes different times.
Scott Mayers
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Re: The scams of Statistics...

Post by Scott Mayers »

One last point on the nature of fairness with respect to nature is that if nature is determined, than in reality, if a perfect random machine operates in it, it too would have to act determinately (as in predictable with respect to it). This would require it to cycle through each optional toss or dice throw in a way that gives each option its fairness. In contrast, if it allows for indetermination, than all we could trust is a type of bell curve for measuring results over actual experiments. Since we cannot determine whether nature is perfectly ONLY determinate NOR indeterminate, since it could be both, than we have to consider all possible options as viable. Thus, the 1/2 solution AND the 2/3 solution are both true but we cannot determine where these occur.

So, the Host can decide to select always the left option in a deterministic way when he has options to show both goats OR he can flip this to always choose the right door, OR he can flip a coin to let it decide instead for him.
mickthinks
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Re: The scams of Statistics...

Post by mickthinks »

Scott, the meaning of your brackets is unclear and leading you into confusion, I think.

Code: Select all

1st pick	Monty tosses a coin	2nd pick	probabilty
				and reveals
				
						Goat 2		-	 Car		   1/6	
  Goat 1 <
						[Car]		-	  [Goat 2]		   [1/6]
 			 	
 			 							 
						Goat 1		-	 Car		   1/6	
  Goat 2 <
						[Car]		-	 [Goat 1]		   [1/6]	
 			 
 			 
						Goat 1		-	 Goat 2	   1/6
  Car	 <	 							 
						Goat 2		-	 Goat 1	   1/6
What I bracketed above can be understood as what option is NOT actually available but counts if we have to 'fake' the toss in the host's mind for those when the Guest selects any goat. That is, the options above are the 'real' options even though the Host could not select the ones in the brackets. ...[/color]

If the brackets around "Car" and "Goat 2" in the second row of your table are meant to indicate something that can't happen, then surely the brackets around "1/6" indicate that 1/6 cannot be the probability. So [1/6] = 0, and your table accounts for an incomplete 4/6 of the total possibilities. You say that these two bracketed options are not actually available. so to complete your table you need to include descriptions of what Monty actually does instead. For example, when the contestant first picks the door with goat 1 behind it and we fake Monty mentally tossing a coin which indicates he should now open the door with the car behind it, which is the door-opening option not available to him at that point, what does he do instead? He opens the door with goat 2 behind it. He does this 1/6th of the time, in addition to the other 1/6 of the time he opens the door with goat 2 behind it after his fake mental coin-toss indicates that as the door he should open.

So the contestant's chance of first picking goat 1 and then winning the car by switching to the remaining closed door is not 1/6, as given in your table, but 1/6 + 1/6 = 1/3, as given in mine. In the same way, for the same reasons, the contestant's chance of first picking goat 2 and then winning the car by switching to the remaining closed door is not 1/6, as given in your table, but 1/6 + 1/6 = 1/3, as given in mine. This means that the contestant's total chance of winning the car by switching to the remaining closed door whichever door he picks first, is not 1/3, as given in your table, but 1/3 + 1/3 = 2/3, as given in mine.

In contrast, if it allows for indetermination, than all we could trust is a type of bell curve for measuring results over actual experiments. Yes, of course. That is how experimental random distributions work out. You make it sound like this is an obviously unacceptable conclusion. Do you prefer to believe in a completely deterministic universe, in which there are no random outcomes?
Scott Mayers
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Re: The scams of Statistics...

Post by Scott Mayers »

mickthinks wrote:Scott, the meaning of your brackets is unclear and leading you into confusion, I think.
The 1/6 would be invalidated precisely for what you said. I was showing how if we treat all six real possibilities without respect to the ones in brackets to be removed or accounted correctly, they lead to four possibilities only to deal with. So it would have to be 'corrected' to reflect this. So you'd have to subtract the two sixths (=1/3) from the 2/3 solution = 1/3 if accepting the initial part.

I already reintroduced this topic elsewhere and it got heated there with increasing insults or endless misinterpretations by the majority of those commenting there rather than to deal with the points I was making. I know what I'm talking about and will come back to it but need a break. I disagree to the faith that many stereotypically place into statistics because of how in the details often suggest some deception of one sort or another. It's like how a magician might be falsely interpreted as ACTUALLY doing something truly miraculously. To the magician, they know their presentation is a trick just as the proponents of puzzles (and many paradoxes) intend to show an unusual perspective leads to odd results. What I wanted to show is how both the perceptions of the magician AND the audience are valid as it is with statistical puzzles like this one to which I thought might be understood.

Yet most think there is simply ONE solution and one perspective that is ever valid. So when I say that 1/2 is also a valid solution, others who have learned the 'trick' transfer their own bias to the perspective of the magician, and see the 2/3 solution only. But while the crowds get too confident that they've understood the magician's tricks, like when those like Penn and Teller do as they supposedly expose the secrets of their own trade, they forget that the magician is counting on just such misdirection by giving their audience that confidence.

The proponents of the puzzle have derived this from past sincere logical inspections of paradoxes who have advanced an argument based on a problem of zero and infinities in math, logic, and science. I thought that I'd tackle this one to show how their are sincere REAL multiple solutions. For instance, if you only do the Monty Hall Game once, odds actually have no meaning to the result. You either win 100% or you do not win 100% in the end of the game. Note that you don't 'lose' in reality unless it actually costs you to play, another factor which gets missed in a proper measure of probability.
In contrast, if it allows for indetermination, than all we could trust is a type of bell curve for measuring results over actual experiments. Yes, of course. That is how experimental random distributions work out. You make it sound like this is an obviously unacceptable conclusion. Do you prefer to believe in a completely deterministic universe, in which there are no random outcomes?
The trick is about the zeros involved that can be interpreted in multiple ways. AND, the significance of using probability itself ALWAYS deals with indeterminate factors to which you discover that by taking one determined position, you lose an ability to determine some other factor involved in the same problem. This is what lead to the "Uncertainty Principle" in QM. And yet while the logic of this has been clearly presented there, people are STILL fooled into thinking the oddities in QM are saying something sincerely paradoxical in nature itself. And I can agree. But I see the perspectives and know why they appear this way. This puzzle is a good example.

But I'm already investing in this when I needed a break from it. :? I'm presently determining the precise logic of Godel's Incompleteness Theorem, Turing's equivalent in computation, and others as it relates to this as well as to all the other interconnected philosophies on logic with respect to math and science of the early 20th Century. I hope this at least helps in some way.
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