[sorry, resending because I hadn't realized this also went to the list]
I suck at explaining basic probability to a 7 year old. You want me to
try *Markov Chains*?
While I agree that the Galton Board is much more transparent (which
is why it was the first suggestion) keep in mind that you don't have
to explain the theory behind them or actually calculate any
properties, just stick with simple observations. [Those, like Deepa,
who wish to see the concrete observations first please skip to the
bottom of this post]
I was thinking they might be nice because you have a choice of source
text (her favorite stories or poems, something with which she's
familiar), they sometimes produce entertaining output, and it's
possible to observe that:
- one never observes probability 0 transitions
- one always observes deterministic (probability 1) transitions
- and otherwise, high probability transitions are generally more
common than low ...
If those concepts stick, then you might eventually move to more
advanced stuff:
- having a probability 0 event AND any other event, even probability
1, still never happens
- having a probability 1 event OR any other event, even probability
0, always happens (when the antecedent to the transition occurs)
- the conjunction of several high probability events is not as likely
- the disjunction of several low probability events is more likely
Or possibly even (after several years?):
- for suitable inputs (those with deterministic tails) there is
absorption: the output is non-deterministic only until encountering
the tail, at which point it becomes deterministic.
- as a long output text gets longer, although transition ratios tend
towards the basic probability ratios, transition differences usually
increase.
And even more subtly:
- if a transition occurs in the output, you know it definitely
occurred in the input (but not vice versa)
- if a transition doesn't occur in the input, you know it definitely
won't occur in the output (but not vice versa)
this kind of "lossy inverse" (where an equality is replaced by a pair
of inequalities) occurs frequently in mathematics and often signals
there's an abstraction to be found (like a pair of data structures
that don't quite round-trip; the values where they *do* round-trip
are the fixed points so beloved of computer science)
-Dave
:: :: ::
Now for a concrete example. I've given the last Silk Digest to Mark
V. Shaney, who, after some reflection, has responded:
Obviously, if the event in question happened, or else is included
in the lottery, and that probability is always calculated by the
total number of possible outcomes, this contradiction is explained,
and empirically valid with your perceived paradox with conventional
frequency based probability.
Given one desired outcome (rolling a 1) and two potential outcomes
(rolling or not rolling a one is important as well.
The probability that the roll will result in a system where
probability is always calculated by the total number of possible
outcomes, this contradiction is explained, and empirically valid
with your perceived paradox with conventional frequency based
probability.
Given one desired outcome (rolling a 1) the expression becomes 1 in 2.
Charles' example though derives from rolling a red? Your view would
have either red, or not rolling a 2 must also be .5, according to
your interpretation.
But since all possible probabilities must sum to 1, .5(6)=3 (or .5
added for each of the probability of rolling a 1) and two potential
outcomes is the probability of coming up with a 1, and the concept
of probability.
That is the primitive of probability.
To be honest I don't see a simpler way to break it down, maybe
someone else on the list has a better method.
However, I am interested in how you ameliorate the paradox between
the probability is accurately reflected as 1/0.
landon I can TELL myself all this: The probability of rolling a
red, thus 50-50.
and we can see that:
- the transition "thus bottom" has probability 0, and we in fact
never encounter it
- the transition "perceived paradox" has probability 1 (is
deterministic, like "thus 50-50"); whenever we see "perceived" it is
always followed by "paradox".
- the transition "a 1" has higher probability than "a 2", and here
it's true (although not necessarily) that it occurs more often.
- the combination "thus bottom" AND "perceived paradox" doesn't occur
- the combination "perceived paradox" OR "thus bottom" does occur
- fewer sentences have "probability is" AND "of probability" than
either one alone
- more sentences have "probability is" OR "of probability" than
either one alone
- (because this Mark is sentence-structured) as soon as we see
"conventional" we know the sentence must end "frequency based
probability". ("thus 50-50" is a more general absorption, but a
poorer example)
- exercise for the reader: what happens to the transition ratios and
differences as we look at longer and longer subsequences of this text?
- because we see "paradox between" in the output, we know it
definitely occurred in the input. (but even though "six possible"
occurs in the input, we don't find it in the output)
- because "thus bottom" doesn't occur in the input, we know it
definitely won't occur in the output. (but even though "six possible"
doesn't occur in the output, we do find it in the input)
which of the statements above are also true of the following
alternative (also from Mark)?
- Werner Do you have an intuitive problem, because you say that 1
will either happen or occur, or it might not.
The dice idea is nice while Deepa's concept is easier for a child
to understand.
I think that while children need to understand the basic frequency
mathematics, there isn't really any point in bothering at all in
the long run.
In the end, intuition is what we feel is simplest, and that's
Occam's razor.
That's fine, except that it's a rule of thumb not an absolute.
It doesn't prove anything.
Your argument is that flipping that coin will have a six sided die.
You can reduce most events to will or will not happen.
That isn't the true representation of that scenario though.
It can only be .5 if there are n people who participate.
can the ones which are now false be changed (in a "related" manner)
to make them true? what would you need to change to make true
statements about Mark's comments were he to have read all silk-list
messages to date? (what can be said about the likelihood of
transitions in Mark's comments on future silk-list messages?)
:: :: ::
Every morning, the chicken sees the farmer come with food. Until one
day, when he comes with an axe.
