
Isaac Newton as a Probabilist - micaeloliveira
http://fermatslibrary.com/s/isaac-newton-as-a-probabilist
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te_platt
Well, once again my intuition completely fails in a probability problem. My
thinking was the more rolls you make the more uniform your distribution of the
number showing becomes. In the limiting case as number of rolls -> infinity
you are guaranteed to have 1/6 of the rolls be a 6. So the probability of at
least 1/6 is less than 1 for a small number of rolls but approaches 1 the more
rolls you do. Intuition is like a good friend who likes to mess with me at
weird times.

~~~
mturmon
You're on the right track with this idea about the flattening of the
distribution as N grows.

As in the article, let

    
    
      X = # sixes in N rolls
    

and note that the expected value of X is N __*p = N /6 in this case (fair
dice).

The event of interest is:

    
    
      {X >= N/6}
    

We want the probability of this event as a function of N.

It turns out this is a decreasing function of N, and that fact answers the
question ("A is most likely"). But why does it decrease in N?

As you say, the distribution of X flattens out as N increases. The probability
mass is spread out over the numbers 0...N, with a peak at exactly N/6\. (Not
N/6 - 1, or N/6 + 1, but N/6.)

About half the probability mass is between 0...N/6, and half is between
N/6...N. So in general, the probability we care about (A, B, or C) is rather
close to 1/2 (because it is just the right half of the distribution).

Since the distribution is flattening out, as N increases, mass moves out of
the exact center (X = N/6) and toward the left or right. Since the event of
interest is

    
    
      {X >= N/6}
    

and not

    
    
      {X > N/6},
    

the fact that the precise center is steadily losing mass (some moving left,
some moving right) as N increases is critical.

In short: Because the event of interest contains the very center, as more
rolls are done, the overall distribution flattens, and the interval between
[N/6...N] benefits less and less by containing the (ever-more-diluted) center
point.

This is another way to restate Stigler's explanation at the top-right corner
of page 401 in the OP. ("The ranking ... reﬂects the fact that ... P(X = Np)
decreases as N increases and the distribution spreads out.")

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mrcactu5
let me give a ridiculous argument that all 3 events A,B,C are equally likely:

    
    
      * event A involves getting one 6 from six dice
      * event B involves getting two 6's from twelve dice
      * event C involves getting three 6's from eightteen dice
    
      since all three events are in proportion, 
      1:6=2:12=3:18 
      the events themselves must all be equally likely.
    

\--------------------------------------------------------------------------------------------------------------------------------

these faulty arguments must have been thrown out all over the place in
Newton's time... and I doubt many discussions today are more careful.

I can imagine a room full of politicians arguing over the likelihood of a
certain event and three camps emerging as to whether they feel A, B, C is more
likely --- throwing away the correct mathematical argument as too complicated.

\--------------------------------------------

what is the correct argument anyway? For the third event

    
    
      C(18,3)*(1/6)^3(5/6)^(18-3)
    

I believe as we put larger and larger numbers we do get a limit, so these
numbers are tending to zero:

    
    
      C(6*k,k)*(1/6)^k(5/6)^(5k) ---> 1/sqrt(2*pi*(5/6)*k) ---> 0
    

so that event A could be the most likely. This does not sound like Isaac
Newton at all.. instead more like Jacob Bernoulli

~~~
JadeNB
For what it's worth, the article formalises "one 6 from 6 dice" as " _at
least_ one 6", and similarly for the other events, which throws off your
probability (which works with " _exactly_ one 6") and hence your asymptotics.

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dpflan
Here is the Newton-Pepys problem explained by Professor Joe Blitzstein in the
Harvard class Stats110:
[https://www.youtube.com/watch?v=P7NE4WF8j-Q&feature=youtu.be...](https://www.youtube.com/watch?v=P7NE4WF8j-Q&feature=youtu.be&t=17m47s)

We see that probability is counterintuitive and complete understanding can
elude the strongest of minds.

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dschiptsov
Probabilities could be correctly applied only for fully observable and well
defined cases like of an unbiased dice or a fair coin. Everything else is a
misapplication and should be regarded as a type error.

Frequency based approach is the right tool, but it could explain nothing about
the partially observable phenomena, except partial statistics of the outcomes
observed so far.

Attempts to derive any causal relationships from statistics of a partially
observed phenomena which is not fully understood and well-defined is
unscientific and is the source of so many misleading "studies" and
overconfident decisions and policies.

The observations that the sun is raising each day on the east does not tell
anything about whether or not and why it will raise there tomorrow. Only
physics offer a causal explanation.

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tome
Here's my heuristic. I wonder if it can be made rigorous.

Write P_n for the probability of throwing at least n sixes with 6n dice.

1\. P_0 = 1

2\. P_n -> 1/2 as n -> infinity, by the central limit theorem.

3\. I can't see any reason that P_n shouldn't be monononic.

Thus I expect that P_1 >= P_2 >= P_3.

~~~
Chinjut
I had originally heuristically reasoned in a similar but opposite way, to the
wrong answer. To wit:

1\. P_{1/6} = 1/6 [when rolling one die, you have a 1/6 chance of getting more
than 1/6 many sixes]

2\. P_n -> 1/2 as n -> infinity, by the central limit theorem

3\. I couldn't see any reason that P_n shouldn't be strictly monotonic.

Thus I expected that P_1 < P_2 < P_3

~~~
thomasahle
If you calculated P_1 = 0.665 I suppose you would guess P_2 would be between
that and P_infty = 1/2, thus smaller than P_1.

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ykler
There are simple ways to intuit the answer to this problem (simpler than the
"heuristic" argument the author gives), but Newton's argument is amazingly
bad. Surprising he could have thought that way.

