Re: WHY the cards have a memory

On Jan 17, 11:29 pm, Ron Sperber <ronsper...@xxxxxxxxxxxxxxxx> wrote:
Lute wrote:
In my earlier post, I was careful to ask people to set aside their
large-universe bias when considering what I said.  Few, if any of you,
did so.  Indeed, the conventional bias was so immense, that I doubt
that any of you really differentiated between what I WAS saying, and
what I was NOT.

Part of it was my fault.  To be honest, the cards don't literally have
a memory.  That was just a symbolic intro to the underlying
assumptions of probability.  My bad.

I simply said that probabilty theory assumes--- and this is a defining
assumption--- that if you flip a coin (truly randomly), it MUST come
up heads, half the time.  Otherwise, you are not defining fifty
percent.

No, it doesn't. At best, what it says is this. If we keep flipping a
coin over and ever, we expect that the ratio # of heads/# of tosses will
approach 0.5. It never says if I flip a coin 10 times, it "must" come
down heads 5 times. In fact it computes probabilities for anything from
0 heads to 10 heads.
If you flip a coin 10 times, the probability of 0 heads (or 10 heads) is
about .000977, The probability of 1 head (or 9 heads) is 0.0097. The
probability of exactly 5 heads and 5 tails is about .2461. That is if
you flip a coin 10 times, you only expect that about 1/4 of the time
will it be "half the time" heads.



The smartest rebuttal I got was this:

***No. It doesn't.
***it assumes that the probability is heads half the time FROM NOW
ON.
***So, being probable at p<1 it is possible that it won't be 50% over
any span you might name.

Now that is a true statement, but it is somewhat misleading.  Let us
see why:

The definition of the probability of fifty percent is axiomatic.  If
you define fifty percent as anything else, you have contradicted your
initial terms.  If fifty percent is NOT fifty percent, then you have a
pointless paradox.

Probability theory also assumes an infinite number of incidents (an
incident in this case being a coin toss).

But the assumption of infinite instances is, at least in the context
of our known universe, false.  As presently understood, the universe
is finite, closed, and unbounded.  That understanding is already being
seriously challenged (see M-theory), but for the moment, it is
workable.

Therefore, in any universe, where the laws of probability are in
effect, whether infinite or finite, probability assumes that if X has
a fifty percent likelihood, it WILL indeed occur fifty percent of the
time.

Don't make the mistake of thinking that a definition of probability (a
purely mathematical concept) is governed by physical reality. It isn't.
It is a model of events whose outcomes are random.





Again, this is a statement of definition

Why is this important?  Because according to those who believe that
literally anything is possible, it is literally possible for universes
to exist (if there are an inifinite number of them), in which every
single coin flip is zero.  Forever.

Imagine the "people" who live in such a universe.  They could spend
aeons in their futile, scientific quest for the universal principle
that REQUIRES every coin toss to be heads.

And if there are an infinite number of universes, then there MUST BE
an infinite number of them in which the most minuscule probable
outcomes must ALWAYS occur.

Indeed, our universe could itself be plausibly described as the
proverbial explosion in the printshop, resulting in the complete works
of Shakespeare.  We could plausibly be nothing more than a continuin
series of unlikely, random events, giving rise to the illusion of
natural laws and logical principles.  And such a universe might, at
any instant, revert to a bad Monty Python movie, or worse yet, to a
good Monty Python movie.

And if that is the case, then there is no ultimate orderly basis of
reality, but only a complex dice game in which ANYTHING can happen
(and eventually will).  Science requires that the laws of nature be
orderly, or else, there is no point in seeking out natural principles.

The laws of probability may not be as we understand them to be.  But
they are my starting point for this discussion.

The laws of probability aren't a physical thing. They are axiomatic
mathematical laws. So they are defined, NOT empirically determined. They
do serve as a pretty good model for events that are essentially random.
(coin tosses, shuffles of a deck of cards, rolling dice, etc.). But
saying an event has p=0.5 just says that we expect the percentage of the
time something occurs to get closer to 0.5 the more times we perform the
experiment. Lets say we flip a coin 500 times and we get 300 heads.
Probability theory doesn't imply that we "catch up" with more tails
later on. But lets say we flip the same coin 1000000 times afterwards
and we get 500000 heads, 500000 tails. Now the percentage of heads in
the combination is .500049975. So those extra 50 heads compared to what
was expected in 500 tosses becomes insignificant in over 1000000 tosses.



Nice post. My friend once postured some people are less lucky than
others. I like that line to demonstrate that getting lucky a few more
times than expected is not what makes some players winners.

(He doesn't win).
.

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