Does a Probability of 1 Really mean a Dead Cert

[Stephen Tashi]

Once that is done then my argument applies.

According your agument,if we are drawing numbers from a uniform distribution on [0,1], I can remove the possibility of drawing $\pi/4$ since it has probability 0. If I proceed to remove all events that have zero probability, there won't be any events left.

I'm not saying this mathematically invalidates your conclusion in a practical sense because the axioms of probability theory don't say whether you can or cannot take random samples from a uniform distribution in the first place.

I think any practical implementation of sampling amounts to sampling from a discrete random variable with a finite number of values. I don't think it is controversial to say that events with probability 1 in such probability spaces always happen if the space correctly models the events. (For example, if a coin has a possibility of landing on its edge and your probability space only includes the events "heads" and "tails" then you have a wrong model. )

When we consider spaces with infinite outcomes (such as space of infinite sequences of coin tosses or random draws from a uniformly distributed random variable) then whether events with probability 1 always happen cannot be tested by practical methods. (It's an interesting question whether Nature herself can take samples from such distributions.)

There are certain questions in mathematics that are "undecided". But to be any kind of question, it must be precise. For example if a statement about all groups is "undecided" then the statement is precise enough that you can look at some particular group and see if the statement is true or false about it. The question of whether whether an event with probability 1 is a "dead certainty" is not an undecided question in probability theory. It isn't even a question at all! It is not precise enough, within the terminology of probability theory, to have a specific meaning.

There is a theory called "possiblity theory" that uses the terminology of "possible" and "necessary" events. I don't know if people have worked on combining it with probability theory.

 

[atyy]

I don't think it is controversial to say that events with probability 1 in such probability spaces always happen if the space correctly models the events. (For example, if a coin has a possibility of landing on its edge and your probability space only includes the events "heads" and "tails" then you have a wrong model.

I think Jano L's argument is that we should say it the other way: if an event is certain, it is correctly modeled as having probability 1. (But not that if an event has probability 1, then it is certain.)

For the case of an event that is modeled as having probability zero, but occurred once, maybe the practical way to decide whether to reject the model is some "traditional way" like the chi-squared test? I haven't tried it out yet.

 

[bhobba]

According your agument,if we are drawing numbers from a uniform distribution on [0,1], I can remove the possibility of drawing $\pi/4$ since it has probability 0. If I proceed to remove all events that have zero probability, there won't be any events left.

Indeed continuous event spaces are problematical - I think a rigorous development along the lines of my argument would need some kind of limit procedures and the introduction of distribution theory - here I mean distributions in the Schwartz sense ie the Dirac Delta Function etc.

Thanks
Bill

 

[atyy]

I attempted to run a chi-squared test with

Category A, observed = 1, expected = 0
Category B, observed =10000000000, expected = 10000000000

I got as the result:
The chi-square test is not possible when any of the expected values are zero.

Which supports the interpretation that if an event with probability 0 occurs, then the model is inadequate (but not because it fails the hypothesis testing).