At that point the only apparent mystery that remains is why when you throw the die only 100 times, most of the time each side shows up about 1/6 of the time, and as I described in my previous post that can be explained with statistics, there is no need to invoke any fundamental randomness. — leo
but it is important to see that in some rare cases, even if you pick the initial conditions as randomly as you can, you can still get frequencies that are totally different from the theoretical probability (for instance getting the number three 1000 times in a row even though you have thrown the die in many different ways without knowing the outcome in advance, this is very rare but it can happen). — leo
At that point the only apparent mystery that remains is why when you throw the die only 100 times, most of the time each side shows up about 1/6 of the time, and as I described in my previous post that can be explained with statistics — leo
They are guesses and we guess because we are ignorant. — Harry Hindu
You seem to be saying that probability = ignorance but that would imply that there is no such thing as randomness or even chance.
If that's the case then consider:
1. A theoretical probability assumes randomness in its calculations. The theoretical probability for a three is 1/6
2. The die thrown 1 million times will show a three 1/6 of the 1 million throws
2 is exactly as predicted by 1 and 1 assumes randomness.
According to your claim then our ignorance led to the random behavior of the coin? How is this possible? How can my ignorance lead to randomness? — TheMadFool
There is such a thing as randomness and chance. They are ideas that stem from our ignorance. Like every other idea, they have causal power. It's just that you are projecting your ignorance/randomness/chance out onto the world where their only existence is in you head as ideas. — Harry Hindu
How does my ignorance cause the die to become random?
Separately, I must ask you this:
Are all random and chance events caused by our ignorance? — TheMadFool
Your ignorance doesn't cause the dice to do anything. Your ignorance causes you to think of the world as probabilities and chances. — Harry Hindu
If it was all probability, then how is it not probable that you roll a 10 on a six-sided die? What constrains the possible outcomes? — Harry Hindu
So you're right that "unexpected" outcomes such as 20 threes in a row can occur in a 100 throws of the die. However, as the number of experiments are increased, say to a million throws, the frequency of threes in that million will be approx. 1/6. — TheMadFool
so if you always throw the die in exactly the same way you always get the same result. — leo
However if there are N different ways to throw the die (say 6 gazillion ways), and you throw the die once in each way, and the die is perfectly symmetrical, you will indeed get each side with frequency 1/6. — leo
The law of large numbers does not explain why if we don’t explain why that law works. — leo
It follows from the law of large numbers that the empirical probability of success in a series of Bernoulli trials will converge to the theoretical probability. — Wikipedia
Now let’s say there are 1 gazillion different initial conditions that yield the outcome three. That means you can throw the die 1 gazillion times in 1 gazillion different ways and always get the outcome three — leo
5. Now imagine you throw the die without looking at which initial state the die achieves. You will see the familiar result that each outcome is 1/6 of the total number of times the die is thrown. This concurs with increasing accuracy the greater the number of experiments that are performed. — TheMadFool
9. Somewhere in the chain events, randomness was introduced into the system. The only place possible is at the time you put the die in one of the six initial states and this was random. This makes complete sense when you consider what you said: — TheMadFool
I'd like to give my own "solution" to the paradox:
A deterministic system can't be random and the die is behaving as if it is random. This implies that a random element was introduced into the system — TheMadFool
It doesn't imply that. A number-generating algorithm simulating 1000 throws can be completely deterministic — Andrew M
. Yet all the outcomes taken together would follow a probability distribution — Andrew M
If you don’t look at the initial state, you may pick unwittingly the same initial state every time (or a member of the set of initial states that yield the same outcome) — leo
However I don’t agree that there is a fundamental randomness that is introduced. — leo
The basic theory of probability and statistics are pretty well established, but when it comes to practical applications there are some problems. Remember when vitamin E was popular for one's health? Then another study showed it had a negative impact on health. Vitamin C is great for this and that, then suddenly it wasn't.
The medical profession does not have a sterling reputation for statistical studies. Sometimes this is due to multiple experiments in which outliers are given undue consideration. Sometimes the experiments are poorly designed.
However, having said this I will tell you that probability theory still leaves me a little uneasy, even though its applications have been largely very successful. Probability waves in QM? Who'd have thunk? :brow: — John Gill
Because...
we can introduce randomness or more accurately pseudo-randomness into a deterministic system. — TheMadFool
pseudorandom — Andrew M
I wonder how one differentiates the true random from pesudorandom? — TheMadFool
This is exactly what bothers me. It should be possible to bias the experiment towards a particular outcome. Yet this doesn't happen and the die behaves in a completely random fashion as is evidenced by the frequency of outcomes in an experiment of large enough number. Why? — TheMadFool
Is there something that still isn’t clear? — leo
Randomness and chance would just be terms indicating one's ignorance of the relevant information for making correct predictions. — Andrew M
Perhaps I don't see the relevance of what you're saying to what is a actually bothering me. Kindly read below. — TheMadFool
I've given it some thought and I think you both are correct but not in the way you think. — TheMadFool
2. Non-deterministic or probabilistic patterns. A die throw is effectively random but any sufficiently large experiment will demonstrate that the outcomes have a pattern viz. that three appears 1/6 of the time, an odd numbered face will appear 3/6 of the time. — TheMadFool
Also bear in mind that a deterministic pattern will differ markedly from a non-deterministic/probabilistic pattern. The latter will exhibit multiplicity of outcomes will the former has only one determined outcome. — TheMadFool
As I keep telling you again and again and again, sometimes no matter how large your experiment is, it doesn't exhibit the pattern you mention. Sometimes you might throw the die 1 billion billion billion billion billion billion billion billion times and always get the same number, or never get some number. It is extremely rare, that's the only reason why you haven't noticed it. — leo
This is wrong also, you throw the die in different ways that's why there is a multiplicity of outcomes, otherwise what you're saying would imply that if the die behaves deterministically it would always land on the same side no matter how we throw it, THAT would be the weird thing. — leo
Just to be clear, deterministic system A is one single die roll. The next die roll would be deterministic system B, and so on.Imagine a deterministic system A (a fair die with 6 sides). Once we have all the information on A we can make accurate predictions of how A will evolve. Deterministic systems will have specific outcomes right? There's nothing random in A and so however A evolves, everything in A will show a pattern and there won't be any variation in the pattern. — TheMadFool
We may know the formula for gravity, which tells us how two massive bodies will interact via gravity, but we still need to know the mass and distance between the two objects in order to predict what will happen over time. We still need to have those conditions plugged into the formula.Please note that patterns are of two types which are:
1. Deterministic patterns. A good example would be gravity - there's a force and that force acts in a predictable manner.
2. Non-deterministic or probabilistic patterns. A die throw is effectively random but any sufficiently large experiment will demonstrate that the outcomes have a pattern viz. that three appears 1/6 of the time, an odd numbered face will appear 3/6 of the time.
Also bear in mind that a deterministic pattern will differ markedly from a non-deterministic/probabilistic pattern. The latter will exhibit multiplicity of outcomes will the former has only one determined outcome.
Imagine now that we lack information i.e. we're ignorant of factors that affect how A will evolve. We assumed A to be deterministic and given that our ignorance has no causal import as far as the system A is concerned, system A should have a deterministic pattern. However, what actually happens is system A now exhibits a non-deterministic/probabilistic pattern.
I will concede that there was a lack of information about system and that is ignorance but that has no causal import on A which should be exhibiting a deterministic pattern because system A is deterministic as we agreed. However, the actual reality when we do experiments we observe non-deterministic/probabilistic patterns. — TheMadFool
Deterministic systems can behave probabilistically
Ignorance or rather the impossibility of knowing was the actual impetus for the development of probability theory — TheMadFool
It seems to me you believe you have understood while you haven’t really understood. — leo
You're conflating knowing one die roll with knowing all of them — Harry Hindu
Tell me what is it that I didn't understand. — TheMadFool
You believe that the die behaves non-deterministically, that’s wrong. — leo
Well, what is the best way to model a die throw in your view?
1. Probability
2. Determinism
Both right? — TheMadFool
I think you’re conflating probability and non-determinism. — leo
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