When I said "it's assured", I meant that someone assures you. Let
me start over:
Say there are going to be some events whose outcomes can't be predicted
in advance. For some of those events, you intend to bet that a certain
outcome will happen. You declare what ourcomes you'd bet on in the
events. All the events except for the 1st one are all the same, making
it easy to specify your wagered outcome for all the bets that you might
make.
Someone who knows more about those events than you do says, correctly,
"Those events all have in common a certain property that allows me to
assure you that if you bet on enough of the events, using a certain
same payoff ratio specified by me, betting on the outcomes that you
indicated, then you can make your overall gain
as close as possible to zero (as close as possible to no overall loss
and no overall gain), merely by betting on sufficiently many of the
events." The person specifies one value of the payoff ratio to use for
all of your bets.
The payoff ratio, r, of a bet is the amount that you win if you win,
divided by the amount that you lose if you lose.
Let P = 1/(r+1).
Say there will only be one event. Your predicted outcome's probability is
the value that
P would have if that event were combined with a large number of
events, identical to eachother (maybe drawing numbers from a bag, etc.),
and someone makes the assurance that is in quotes above, about all the bets
you make, after you've indicated to him how you'll bet on the 1st event and
on the subsequent identical ones.
[end of definition]
I could have just said that that person assures you that you can make
the fraction of successes as close as possible to a certain value, by
betting on sufficiently many of the events. But I initially had a more
ambitious intention of letting r & P vary from one event to the next.
That would be better, but it seemed more difficult, and I didn't have
it ready in time. But that's why I used the approach that I used.
Another version, in which that assurance about the fraction of successes
replaces the assurance about the overall gain, can be considered
version 2 of my definition.
Now, Martin said incorrectly that I mean something different by "if" than
mathematicians do, and then said that, for that reason, what I mean by
"probability" isn't probability. So Martin, is what I call probability, in
the 2 versions of my definition, not probability?
Or were you maybe making some careless statements--a wrong one, and
a premature one?
It goes without saying that, typically, it wouldn't be possible for
someone to correctly make the assurance that is made in my definition.
My definition uses a series of events and declared betting intentions
for which that assurance can be made.
Whether or not it's now written as precisely as it could be, my
definition shows what I mean by probability. Even if it would need
rewriting to make it precise, it still shows what I mean, in its present
form.
Mike Ossipoff
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