probability

If probability is based on empirical assumptions , how valid is it ?

Originally posted by kaminsky
If probability is based on empirical assumptions , how valid is it ?

It depends on the context. As long as the empirical assumptions are fully stated, the probability calculation itself is entirely valid. If the assumptions turn out to be wrong, then the results too will be wrong.
If the die is weighted, don't expect the statistically predicted results for an unweighted die.

Originally posted by kaminsky
If probability is based on empirical assumptions , how valid is it ?

I interpret "valid" to mean internally consistent, that is a correct application of reasoning.

Maybe however you mean - what weight can I attach to a probability when I know that it is based on empirical assumptions? To some degree that depends what weight you are willing to attach to the assumptions, which ought to be stated. Anyway, more likely your question will arise when dealing with findings from a "scientific" study of some kind, and then a lot of questions have to be asked about methodology - the way in which the study is constructed in order to support the resulting probability statement.

For example in psychology a standard assumption is that, for any parameter that can be measured on a continuous scale, the findings for the general population will conform to a normal curve. This is not always the case but even of it is a reasonable assumption, then problems arise ensuring that the subjects on whom we carry out any study are representative of the general population and not a skewed sample (systematically excluding or over-representing segments of the population). For instance, studies that are done using students as subjects are highly likely to give a seriously skewed picture compared with a true sample of subjects from the general population. This may matter more for some types of study than for others. A lot of studies are also dubious because the sample size is far too small to carry the weight. This may be especially so with health issues and wild claims based on tiny studies.

Problems also arise interpreting the conclusions from a statistical study. It is important to be clear just what we mean by any probability statement and in particular to appreciate the significance of sampling error. The classic problem is surely that one study which gives evidence of something interesting, however well conducted and reported, may fail to be replicated. There are countless health issues, and matters of social importance, in which people refuse to recognise the need for any finding to be replicated before giving it undue weight.

One study that says all chess players on RHP are fascinating and charming will surely not be enough to convince us, and we will want to see that replicated before we use the site as a dating agency. It may give a misleading result because it is based only on members of the Ireland Clan.

Originally posted by finnegan
I interpret "valid" to mean internally consistent, that is a correct application of reasoning.

Maybe however you mean - what weight can I attach to a probability when I know that it is based on empirical assumptions? To some degree that depends what weight you are willing to attach to the assumptions, which ought to be stated. Anyway, more likely your ...[text shortened]... cy. It may give a misleading result because it is based only on members of the Ireland Clan.

Do you let in North Irish?

Originally posted by AThousandYoung
Do you let in North Irish?

Only if they can write either English or Irish sentences I can understand. I'd obviously prefer an Irish sentence I could understand.

Originally posted by kaminsky
If probability is based on empirical assumptions , how valid is it ?

There are two kinds of probability: empirical probability and theoretical probability. The theoretical probability of rolling a die and getting a 5 is 1/6.

On the other hand if you are not sure the die is "fair" (i.e. not loaded), you might roll the die 1000 times and tally up the number of times you get a 5. Say you get a 5 a total of 158 times. Then the empirical probability of getting a 5 is calculated as 158/1000, or 0.158. That is, empirical probabilities are based on observation and data.

Both empirical and theoretical probability assumptions seem flawed since they rely on inductive logic , because something has happened before, it will happen again. Im not a mathmatician, but it does not seem very mathmatical to rely on assumed observations of events . The internal maths of probability maybe faultless, its the first principles I'm questioning.

Originally posted by kaminsky
Im not a mathmatician, but it does not seem very mathmatical to rely on assumed observations of events.

That's why that part of it is stated as an assumption and is not part of the mathematics.

The internal maths of probability maybe faultless, its the first principles I'm questioning.
But the principle that because something has happened before it will happen again has nothing to do with probability. It is a basic principle of existence and without it we are lost. If we do not assume that the laws of physics will be the same tomorrow as they are today, then we can predict nothing - not even whether or not the sun will rise tomorrow.
But we can surely at least say that in the past, the universe operated via a set of rules.

You would not have typed your post unless you assumed that someone would read it - you made the assumption.

Originally posted by kaminsky
Both empirical and theoretical probability assumptions seem flawed since they rely on inductive logic , because something has happened before, it will happen again. Im not a mathmatician, but it does not seem very mathmatical to rely on assumed observations of events . The internal maths of probability maybe faultless, its the first principles I'm questioning.

Try explaining mathematics in terms of logic.

Originally posted by KazetNagorra
Try explaining mathematics in terms of logic.

...reaching for Principia Mathematica... I may be some time...

On a philosophical level I understand this project collapses in the face of Godel, but not without laying the foundations on which it became possible for computers to interpret logical statements in terms of 0 and 1 (viz mathematical logic) which in turn translate into an electrical on/off switch. From this I assume that (like the Bumble Bee supposedly being physically unable to fly, yet it does,) so also it may be impossible to reduce Reason to numbers, yet we put this into practice every day.

That's another impossible thing I believe before breakfast!

Originally posted by kaminsky
Both empirical and theoretical probability assumptions seem flawed since they rely on inductive logic , because something has happened before, it will happen again. Im not a mathmatician, but it does not seem very mathmatical to rely on assumed observations of events . The internal maths of probability maybe faultless, its the first principles I'm questioning.

Wha...?

Empirical probability of an event E is defined to be the number of times E is observed to occur divided by the total number of observations. There is no flaw there, because it's a definition. So if 238 individuals are recorded to be struck by lightning in the U.S. in 2011, then the empirical probability of someone getting struck by lightning is determined to be 238 divided by the population of the U.S. Empirical probabilities, however, are only as good as your data and will (like polls, surveys, and the like) tend to give somewhat varying values as more data is collected.

If n(E) is the number of outcomes of an experiment that are favorable to an event E and n(S) is the total number of possible outcomes of the experiment (also known as the sample space), then the theoretical probability of E is defined to be given by n(E)/n(S). So, with the roll of a die, the number of outcomes favorable to rolling a five is 1, while the total number of outcomes of the experiment is 6 (i.e. six different numbers could turn up). Hence, the theoretical probability of rolling a five is 1/6.

The only thing that can really get tricky about probabilities -- and in particular theoretical probabilities -- is that they can depend on the amount of information an experimenter knows. A process that appears unpredictable may in fact be merely misunderstood, and additional knowledge about how the process works may alter the probabilities of various events occurring. But that's not a fallacy. It again comes down to what you know about a process.

I'm assuming your problem is with assumptions that are made to compute the probabilities of a physical process behaving in a certain way. That's certainly a problem for physicists and other researching scientists on the field, but it is not a crisis for the mathematical discipline of probability theory itself since it is a conceptual (i.e. non-physical) construct based on a few basic definitions and axioms. As a system, probability theory is proven to be logically consistent.

If I can put my question in some context , it might help since Im not mathmatician and it might not be clear what I mean . A visit to the London science museum ,where an exhibit about ERNIE ,the random number generator ,states that the numbers generated could actualy never be truly random,coincides with me reading Einsteins statement that god does'nt play dice with the universe. This leads me to discuss ,while drinking with my brother wether randomness really exists and if it does'nt what are the consequences.It seems to me that probability theory seems to imply randomness ,which obviously is a problem for probabilty theory ,because of the mismatch between the real world ,where all events are weighted (non random) and the abstract world ,where randoness can exist.
It appears to me that that the first principles of say arithmetic , where oneness and twoness seem to exist in both the real and abstract world ,has a sound footing ,while probability may require a leap of faith . The maths in probability is developed in the abstract world, then tested experimentally in the real world to see if it fits.
I hope this makes things more clear .

Originally posted by kaminsky
This leads me to discuss ,while drinking with my brother

<voice=Adam Savage>
Well, there's your problem!
</voice>

Richard

LOL your probably right.

Originally posted by kaminsky
If I can put my question in some context , it might help since Im not mathmatician and it might not be clear what I mean . A visit to the London science museum ,where an exhibit about ERNIE ,the random number generator ,states that the numbers generated could actualy never be truly random,coincides with me reading Einsteins statement that god does'nt play ed experimentally in the real world to see if it fits.
I hope this makes things more clear .

A random process is one in which all possible outcomes have equal probability of occurring. There is such a thing, at least conceptually. Like tossing a coin or rolling a die. Though, if a coin or die is not constructed "perfectly," it will tend to favor one side over others.