[EM] Clarification on Bart's Pij definition

[MIKE OSSIPOFF]

Martin Harper said:

>probability is
>defined using mathematical ifs - and if you calculate it some other way,
>then you won't get a probability. If you don't get a probability, you
>can't use it to accurately calculate the utilities of voting for or
>against each candidate.

Wow. I'm glad that someone finally pointed out to me that all these
years I've been ignorantly talking about probabilities, without knowing
what probability is, because I don't know what "if" means.

And it's a good thing that I didn't actually use my wrong non-probabilities, 
based on a wrong "if",  to determine how to vote in an actual election.

But Martin, since you've brought this up, and said that "if" and
"probability" mean something very different from what I've believed
all these years that they mean, shouldn't you tell me what they
actually mean?

1. What is the mathematical meaning of "if"?
2. What do you take to be the definition of probability?

I should say what my wrong definitions of "if" & "probability" are,
since I've been using those terms here all this time. Of course not
only do I not know the genuine mathematical meanings of those words
(or of no doubt many of the other words in this paragraph, this posting, and 
all my previous postings), but I also am writing these definitions
on short notice. Aside from the fact that my definitions of "if" &
"probability" are wrong, non-mathematical definitions, the short notice
on which I'm writing them could result in their not even being as
good as they could have been written, even by a nonmathematician.

Here's my common-person's definition of "if":

"If A is true then B is true" means that A being true means that it's
certain that B is true".

I freely admit that I don't know the mathematical definitions of "means",
"true", "is", or "that", or what words a mathematician would use instead
of those.

And, if you don't mind my clarifying this, I'm _not_ saying that A
being true causes B to be true.

Now, for a commoner's definition of "probability". When I use "if",
I'm of course using my common people's "if.

Say that on many occasions, some sort of trial takes place. And, before each
trial, you bet, with some payoff ratio (ratio between the money you win if 
you're right and the money that you bet), that a certain outcome will happen 
in that trial. That payoff ratio is the same for all the trials,
and will be called "r".

Say that your choice of r assures that you can make your overall
gain (positive or negative) as close to zero as you want to by doing
enough trials. P = 1/(r+1) is called the probability that
you'll win the bet in any particular trial.

Of course, in this definition, the win probability, P, is the same in
each trial.

But for any kind of trial, we can suppose that you combine it with many
other trials of some kind, where you are able to accomplish the same overall 
gain by choosing the right r. Of course these would just happen
to be trials that have the same win probability as eachother.

I'm not saying there's a way of determining the right r. I'm only
supposing that there's some r that guarantees that sufficient trials
will make your overall gain as close to zero as you want.

Now, it's late, and I wrote that on short notice. And so I don't
guarantee that it doesn't need touching up. Also, Martin said that a
mathematical definition of probability must contain the word "if", which
is one reason why my probability definition can't be a mathematical
definition of probability. But, if I avoid using the word "if", at
least that means that I'm not using it with a mathematically incorrect
meaning.

And now that I've given my wrong, common people's, incorrect definitions
of "if" and "probability", I hope that Martin will tell me the
correct mathematical definitions.

Mike Ossipoff
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