[dlorde]

If you can consistently get better than 50%, the coin sequence isn't random. Your choice (prediction) can't affect the coin odds.

[wolfekeeper]

I think people have a lot of trouble with statistics (e.g. the Monty Hall problem is a very classic example), but statistics in general seems to fry people's brains.

Relativity, again, people usually can't hack it.

QM, practically nobody really has much clue!

[Pmb (OP)]

That's not how the game works! If you choose the wrong door, the host is obliged to show you where the prize is by revealing the 98 doors where he knows the prize is not, giving you a sure pointer to the prize. So your chances of winning are 1% if you stand, but 99% if you swap.

Then what I said was caca??

I don't understand. Suppose that the prize is behind door number 2. If I choose door number 1 then he reveals that doors numbers 3-100 have nothing behind them? If so then it seems to me that you have to choose door number 2 after that and have 100% chance of winning.

I don't think I understand that game. In any case I'm not interested. It's getting off topic for me.

[Pmb (OP)]

Can you explain the reasoning here? you surely have the same chance whether you change your number each time or not.

Do two experiments using a single die.

Experiment Number 1: Roll the die 100 times. Every time the number 1 comes up give yourself a penny.

Experiment Number 2: Roll the die 100 times.
On the first roll if the number that comes up is a 1 give yourself a penny.
Roll the die again. If the number that comes up is a 2 give yourself a penny.
Roll the die again. If the number that comes up is a 3 give yourself a penny.
Roll the die again. If the number that comes up is a 4 give yourself a penny.
Roll the die again. If the number that comes up is a 5 give yourself a penny.
Roll the die again. If the number that comes up is a 6 give yourself a penny.
Roll the die again. If the number that comes up is a 1 give yourself a penny.
Roll the die again. If the number that comes up is a 2 give yourself a penny.
(keep doing this until you've rolled the die 100 times)

The probability of you getting more money during experiment number one is greater than that of number two.

My expertiese in combinatorics is too rusty to calculate the excact probabilites. It's been over twenty years since I took that course. Blech!

As I understand it, the only criteria for selecting a lottery number is to avoid one that other people might be likely to pick too; it doesn't help your chances, but if you do win, you're less likely to be sharing the prize.

Never worry about that because its beyond your control and doesn't affect the probability of winning or the amount.

[burning]

The probability of you getting more money during experiment number one is greater than that of number two.

I'm pretty sure that's wrong.

The probability of the number you guessed coming up is 1 in 6 for every roll, regardless of whether you guess the same number each time, a different number each time following a pattern, or a different number each time chosen at random.  The expectation value for the number of wins will then be 100/6 for either experiment.

Can you explain your reasoning why you expect differently?

By the way, I ran the experiments using the random number generator in Excel.  I know that it's not a high quality random number generator, but it should be good enough to imitate a fair die.  I "rolled" 1000 dice at a time and compared the number of wins under the two assumptions.  While I didn't conduct the experiments sufficient times enough to give a conclusive statistical analysis, both methods gave results within a reasonable error range from 1000/6, and neither method showed a tendency to win more often than the other.

[Pmb (OP)]

Can you explain your reasoning why you expect differently?

As I explained above, my combinatorics is very rusty. Think of changing the number is trying to hit a moving target rather than a stationary one. But who knows. I could be wrong. You need to have a solid knowledge of combinatorics to determine this exactly and I haven't done that in decades. So sure, perhaps you're right and I'm wrong. You can always try it and see what happens.

[dlorde]

The probability of you getting more money during experiment number one is greater than that of number two.

I don't see how that's possible. The odds are one in six each time; your choice can't change that. OTOH, if your idea had legs, we could clean out the casinos 

[yor_on]

Is it this you're thinking of Pete?

"If every object in a collection of objects fails to have a certain property, then the probability that a random object chosen from the collection has that property is zero. Turning this around, if the probability that the random object has the property is greater than zero, then this proves the existence of at least one object in the collection that has the property. It doesn't matter if the probability is vanishingly small; any positive probability will do.

Similarly, showing that the probability is (strictly) less than 1 can be used to prove the existence of an object that does not satisfy the prescribed properties.

Another way to use the probabilistic method is by calculating the expected value of some random variable. If it can be shown that the random variable can take on a value less than the expected value, this proves that the random variable can also take on some value greater than the expected value."

"In probabilistic combinatorics, the questions are of the following type: what is the probability of a certain property for a random discrete object, such as a random graph? For instance, what is the average number of triangles in a random graph? Probabilistic methods are also used to determine the existence of combinatorial objects with certain prescribed properties (for which explicit examples might be difficult to find), simply by observing that the probability of randomly selecting an object with those properties is greater than 0. This approach (often referred to as the probabilistic method) proved highly effective in applications to extremal combinatorics and graph theory. A closely related area is the study of finite Markov chains, especially on combinatorial objects. Here again probabilistic tools are used to estimate the mixing time.

Often associated with Paul Erdős, who did the pioneer work on the subject, probabilistic combinatorics was traditionally viewed as a set of tools to study problems in other parts of combinatorics. However, with the growth of applications to analysis of algorithms in computer science, as well as classical probability, additive and probabilistic number theory, the area recently grew to become an independent field of combinatorics."

And this http://www.goldsim.com/Web/Introduction/Probabilistic/MonteCarlo/

"In order to compute the probability distribution of predicted performance, it is necessary to propagate (translate) the input uncertainties into uncertainties in the results. A variety of methods exist for propagating uncertainty.  Monte Carlo simulation is perhaps the most common technique for propagating the uncertainty in the various aspects of a system to the predicted performance.
 
In Monte Carlo simulation, the entire system is simulated a large number (e.g., 1000) of times. Each simulation is equally likely, referred to as a realization of the system. For each realization, all of the uncertain parameters are sampled (i.e., a single random value is selected from the specified distribution describing each parameter). The system is then simulated through time (given the particular set of input parameters) such that the performance of the system can be computed. This results is a large number of separate and independent results, each representing a possible “future” for the system (i.e., one possible path the system may follow through time). The results of the independent system realizations are assembled into probability distributions of possible outcomes. As a result, the outputs are not single values, but probability distributions."

That one sound close to what I called 'patterns' to me. And the number fits too
Anyone that have a simple example of it, maybe?

How using uncertainty to make a guess more certain, without hidden parameters?
Or do I need to assume hidden parameters for it to work? As the game master 'knowing' which door that contained the car, and so never opening that one. (As well as he can't open my first choice of door, as that destroys my later choice, well, as it seems to me?)

[JP]

You don't have to do 100 times to check it.  There's nothing fundamentally different between the way the dice behave after 100 rolls or 2 rolls, and you can substitute 2 sided dice (a coin with numbered faces) for 6 sided dice without changing the fundamentals of the problem.  Your possible outcomes of two flips are:

1,1
1,2
2,1
2,2

and they're all equally likely.

If you choose method 1 (predicting 1,1 as the outcome), your earnings are:
2 cents 25% of the time
1 cent 50% of the time
0 cents 25% of the time

If you choose method 2 (predict 1,2 as the outcome), your earnings are:
2 cents 25% of the time
1 cent 50% of the time
0 cents 25% of the time

And its easy to verify that any guess will have the same odds of winning, since each roll is independent of the others and you have a 50% chance of winning.  This extends straightforwardly to more rolls and 6 sided dice.  Things do change if you just want to guess at the numbers rolled, independent of ordering.  If someone just asks what the two values in 2 flips of this die are, you're best off guessing 1 and 2, since that combination shows up half the time.

[yor_on]

That's pure probability, as I read you JP

But using uncertainty to lower a uncertainty of the other side, aka, not knowing the dices outcome before they show it? Is there really a way to do that? And a example of it please
=

Or better expressed, can you fight the dice or lottery's randomness by introducing your own randomness. It's not the exact same as what Pete suggested but the idea caught my imagination. Is it possible? And in what ways/situations?

[dlorde]

In Monte Carlo simulation, the entire system is simulated a large number (e.g., 1000) of times. Each simulation is equally likely, referred to as a realization of the system. For each realization, all of the uncertain parameters are sampled (i.e., a single random value is selected from the specified distribution describing each parameter). The system is then simulated through time (given the particular set of input parameters) such that the performance of the system can be computed. This results is a large number of separate and independent results, each representing a possible “future” for the system (i.e., one possible path the system may follow through time). The results of the independent system realizations are assembled into probability distributions of possible outcomes. As a result, the outputs are not single values, but probability distributions."

That one sound close to what I called 'patterns' to me. And the number fits too
Anyone that have a simple example of it, maybe?

Weather forecasting. They run numerous projections with a number of models, varying the initial parameters. This gives them a spectrum of possible futures. If the weather is in a reasonably non-chaotic state there will be groups of similar patterns in the result spectrum. The sizes of the groups can be used to give a probability estimate for each predicted weather pattern.

[yor_on]

Nice example dlorde, I reminds me of a 'fractal approach' too, as you look for larger patterns in the patterns visible. It also reminds me of assigning 'weights' in neural networks, as those 'clumps' of patterns closest to each other might be said to be 'weighted up' by probability.

Then we just have randomness left it seems, and Pete's suggestion going the other way defining a ordered approach. Both involves decision making though, even if random in the first case. And those two are the ones I find most difficult to imagine, but rather intriguing.

[dlorde]

Then we just have randomness left it seems, and Pete's suggestion going the other way defining a ordered approach.

There is another major option - chaos. For example, when the weather is in a chaotic state, the simulations come out very different regardless how close the initial parameters are set. It's not random, it's entirely deterministic; but it's totally unpredictable... non-linear dynamics, the Butterfly Effect; it was all the rage n the '80's.

[yor_on]

Yes, I agree. Let me put it this way though, when I think of randomness then I do it from chaos. Maybe that's not correct but to me they become equivalent, although you might want to define randomness to superpositions microscopically, as a example of how I think versus chaos macroscopically. Even though you can call chaos deterministic, as in if we only knew all parameters we could describe it, I personally relate it to a randomness.

Maybe I could express it as I don't think there ever will a possibility of knowing the whole history, of anything. It seems to go through all physics that one, no matter what scales you look at it from?

[JP]

That's pure probability, as I read you JP

But using uncertainty to lower a uncertainty of the other side, aka, not knowing the dices outcome before they show it? Is there really a way to do that? And a example of it please
=

Or better expressed, can you fight the dice or lottery's randomness by introducing your own randomness. It's not the exact same as what Pete suggested but the idea caught my imagination. Is it possible? And in what ways/situations?

No, you can't if the dice are actually random.  You don't have to take my word for it, though.  The 2 coin case is easy enough to think about--you can list out all the possibilities very easily and each flip has a 50/50 chance of producing either 1 or 2. 

[dlorde]

Yes, I agree. Let me put it this way though, when I think of randomness then I do it from chaos. Maybe that's not correct but to me they become equivalent, although you might want to define randomness to superpositions microscopically, as a example of how I think versus chaos macroscopically. Even though you can call chaos deterministic, as in if we only knew all parameters we could describe it, I personally relate it to a randomness.

Well, 'deterministic' means 'non-random' by definition, so perhaps 'pseudo-random' is a better description for the results of chaotic functions.

Maybe I could express it as I don't think there ever will a possibility of knowing the whole history, of anything. It seems to go through all physics that one, no matter what scales you look at it from?

'Having no history' - that could almost pass as a definition of randomness that leaves the question of knowledge open - i.e. can one distinguish a chaotic sequence from a random sequence without knowing the prior history of the system? If I recall correctly, a characteristic of chaotic systems is that islands of order may appear at regular intervals among the apparent randomness, so it should be possible to distinguish randomness from chaos for long enough sequences. How long 'long enough' might be is an open question...

[yor_on]

A intriguing idea dlorde. And yes, assuming Feigenbaums constant one should be able to assume a fractal behavior to chaos giving you mystical attractors and some regularity in that randomness. But then the question becomes how to prove what is random, doesn't it? Haven't thought of that one but it seems to become a mathematical concept defined by what circumstances, degrees of freedom etc something have. In flipping the coin we then have three possibilities, the coin giving us head tail and its edge. Of those we ignore the edge defining it as head and tail, and define it as a 50/50 chance for each throw. The randomness referred to here is not something able to make the coin disappear, it's a mathematical definition of what 'choice' of two it may make landing.

[yor_on]

Now, as far as I remember, although Feigenbaums constant give you a regularity, somethings bifurcations (splits) can not be back-tracked. What does that make it, a random behavior? What is random here? Normally used I would refer to something random as something I can't foresee, meaning its exact outcome. Although I might expect myself able to back-track it after a outcome. But that isn't perfectly correct, is it thinking of that coin throw. Even though I couldn't foresee it, after its outcome I sincerely doubt myself to be able to define exactly why it gave me a 'tail', but, should I be able too? If knowing all parameters?
=

This one is nice http://www.imho.com/grae/chaos/chaos.html
==

Then again, assuming a time symmetry. if I can back-track it I also should be able foresee it, using the same presumption (knowing all parameters). So that one doesn't really say anything. Well, it do state something, if we define it as being of two choices.

Determinism relative indeterminism. Then what we have found so far is not determinism as I think, that as I see no real experimental proof for it. Historically we've used determinism though as a leading star, expecting a linear causality to define the universe, as some clock work. And working from an assumption of the universe wanting it to be as simple as possible, indeterminism must win

[yor_on]

So, either we define it as a 'random behavior' inherent in Chaos equations, or we call them deterministic. If they really are deterministic then we should be able to simplify a chaotic system to the degree where we can control all parameters, and so 'predict' its outcomes. But we can't do that, as far as I know. So that leaves us a randomness, but is it a same randomness as in the coin throws?

Put another way, you're part of a chaotic system, population wise, environmental, Earth itself. As you throw that coin, the coin also must become a parameter in chaos.
=

And what more, well, if one believe in a absolute time symmetry, how do one explain indeterminism? And 'randomness'. I define time symmetry to a logic. I can also use Feigenbaums constant to define a logic, but inside that logic we find randomness and, as I think, also a description of indeterminism. Time symmetry works as a logic because we need it, it gives us a past and and a present and a future. But it's no proof of a arrow able to go backwards, not as long as you use your local clock defining that experiment proving your concept.

[dlorde]

So, either we define it as a 'random behavior' inherent in Chaos equations, or we call them deterministic. If they really are deterministic then we should be able to simplify a chaotic system to the degree where we can control all parameters, and so 'predict' its outcomes. But we can't do that, as far as I know. So that leaves us a randomness, but is it a same randomness as in the coin throws?

As I previously said, chaos is deterministic, not random. It is unpredictable because of sensitive dependence on initial conditions. For example, this means that given the initial parameters, we can't predict on which wing of Lorentz's butterfly we'll find ourselves after n iterations. To discover this, we must iterate the function n times with those initial parameters. In the real world, we can't obtain the initial parameters for chaotic processes with sufficient precision to obtain useful results by iterating the function(s) even if we know it(them). Hence the Monte-carlo simulations for weather, where even for chaotic states there is some hope of identifying the major attractors (although this may mean predicting, say, a 40% chance of stormy, wet weather and a 60% chance of clear, sunny weather!).

The real world has both chaotic and random components. Chaotic because of non-linear dynamic processes, and random because of quantum indeterminacy. The contribution of quantum indeterminacy to chaotic processes in the real world introduces a clouding degree of randomness at small scales, but should be statistically averaged out for macro-scale processes.