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On 13/01/2011 02:27 PM, Invisible wrote:
> Suppose the probability of a catch is unconditionally P. How does the
> length of a typical series of consecutive catches vary as a function of P?
Apparently catching or dropping the diabolo can be regarded as a
Bernoulli trail, and thus performing an open orbit is a kind of
Bernoulli process. In particular, the number of failures before a
success is achieved (or, conversely, the number of successes before the
first failure) follows a geometric distribution.
Specifically, if the probability of catching the diabolo is P, then the
probability of *not* catching it must be 1-P. Apparently the probability
of K successes followed by one failure is
Prob K = P^K * (1-P)
Thus, each series of catches of length K+1 is a factor of P less
probable than a series of length K.
All of this of course assumes that the trails are *independent*, which
they manifestly are not...
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On 13/01/2011 02:51 PM, Invisible wrote:
> Specifically, if the probability of catching the diabolo is P, then the
> probability of *not* catching it must be 1-P. Apparently the probability
> of K successes followed by one failure is
>
> Prob K = P^K * (1-P)
If P is the probability of an event happening, than P^K is the
probability of it happening K times, and 1-P is the probability of it
not happening once. Hence the above formula.
I tried plotting it:
http://tinyurl.com/6bsfnol
As you'd expect, as the catch probability goes up, the probability of
longer chain lengths increases sharply.
Paradoxically, as P becomes very close to 100%, 1-P obviously becomes
extremely close to zero. In other words, /every/ chain length becomes
very unlikely.
If we look not at chains of length K but chains of length /at least/ K,
we get Prob K = P^K. Plotting this:
http://tinyurl.com/5wlkdmf
we discover that indeed, as P increases, the typical chain lengths
increase very sharply. (If Wolfram had bothered to label their axies, I
might even be able to tell you what the probability of 50 catches is. :-P )
> All of this of course assumes that the trails are *independent*, which
> they manifestly are not...
I have no idea how to account for that mathematically, however.
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Invisible <voi### [at] dev null> wrote:
> On 13/01/2011 02:51 PM, Invisible wrote:
> I have no idea how to account for that mathematically, however.
If you dedicated as much time to learning to catch the diablo as you did to
analysing the catch stats, you'd probably be on 501 catches by now ;-)
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On 13/01/2011 03:41 PM, Bill Pragnell wrote:
> If you dedicated as much time to learning to catch the diablo as you did to
> analysing the catch stats, you'd probably be on 501 catches by now ;-)
Plausibly, yes.
From the modelling I've been doing, it looks like once your drop
probability falls below about 10%, small improvements in your catch rate
yield massive improvements in consecutive catch stats.
In other words, the effort required to get from 6 catches to 16 catches
is bigger than the effort to get from 16 to 26. According to simple
mathematical models, at a certain point it should suddenly become easy
to do catch chains of almost limitless size.
I suspect that if you incorporate such factors as limited concentration
time, the reality of the matter is different. ;-) None the less, I'm
excited about the possibilities.
(Now, if only I could get all the *other* stuff right too...)
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Invisible wrote:
> Since then, I haven't managed to get near 50 again, but I have managed
> several teens and a few twenties. For whatever reason, I seem to have
> suddenly made a quantum leap in my skills. O_O
This happens to me a lot. If I practice something for a while, then walk
away for a day, when I come back, the practice seems to have "taken hold" or
something and I'm far better than I was before.
--
Darren New, San Diego CA, USA (PST)
Serving Suggestion:
"Don't serve this any more. It's awful."
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On 13/01/2011 04:13 PM, Darren New wrote:
> Invisible wrote:
>> Since then, I haven't managed to get near 50 again, but I have managed
>> several teens and a few twenties. For whatever reason, I seem to have
>> suddenly made a quantum leap in my skills. O_O
>
> This happens to me a lot. If I practice something for a while, then walk
> away for a day, when I come back, the practice seems to have "taken
> hold" or something and I'm far better than I was before.
Well, I started practising again today, and managed a few 4s and 5s, and
then suddenly just pulled off 51 out of nowhere. Then a 33. And then
down to 10s and 20s for the rest of the session. I have no idea why. I
guess I just figured out some crucial piece of technique, but I have no
idea what!
Well, maybe I do. The throws are smoother now, which probably makes the
catches easier. I'm still having issues with tilt and direction though...
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Le 13/01/2011 17:17, Invisible a écrit :
> And then down to 10s and 20s for the rest of the session. I have no idea
> why. I guess I just figured out some crucial piece of technique, but I
> have no idea what!
The centipede was a very good dancer, making marvel at all kind of
complicated dances, and it was happy.
The naughty and jealous spider ask it: when you do that move, are you
moving foot 95 before or after foot 54 ?
The centipede had no answer, but from that day it was unable to dance as
before, always thinking about the ordering and details. Its dancing
skill was gone.
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> I didn't practise at all yesterday, but I had a go just now. After
> several 4s and 5s, I somehow managed to pull off 51 consecutive catches,
> utterly /annihilating/ my previous personal best of 16. That's over 3x
> better performance!
Yeh it's funny how your body is capable of learning stuff like that,
isn't it? Reminds me of learning to do tricks on my bike, I went
through a similar process of how many times I could bounce on the rear
wheel, at first it seems impossible, then before you know it you can
just go on for as long as you want.
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On 14/01/2011 09:03 AM, Le_Forgeron wrote:
> The centipede was a very good dancer, making marvel at all kind of
> complicated dances, and it was happy.
>
> The naughty and jealous spider ask it: when you do that move, are you
> moving foot 95 before or after foot 54 ?
>
> The centipede had no answer, but from that day it was unable to dance as
> before, always thinking about the ordering and details. Its dancing
> skill was gone.
As a somewhat skilful dancer myself, I can tell you that this effect is
_real_. o_O
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On 14/01/2011 09:50 AM, scott wrote:
>> I didn't practise at all yesterday, but I had a go just now. After
>> several 4s and 5s, I somehow managed to pull off 51 consecutive catches,
>> utterly /annihilating/ my previous personal best of 16. That's over 3x
>> better performance!
>
> Yeh it's funny how your body is capable of learning stuff like that,
> isn't it? Reminds me of learning to do tricks on my bike, I went through
> a similar process of how many times I could bounce on the rear wheel, at
> first it seems impossible, then before you know it you can just go on
> for as long as you want.
Partly it seems to follow from the mathematics of the geometric
distribution. (The number of consecutive successes is not linearly
dependent on success probability; after a certain point, potential chain
length skyrockets.) But partly it's probably something to do with the
inner workings of the human brain. Playing BWV590 used to seem almost
impossibly difficult. Today I do it while not even paying attention. And
it didn't take as long as you might imagine to reach this point.
The high throws are actually easier to catch than the low throws.
Presumably because you have far more time to react. Indeed, the hardest
part of catching a high throw is having the nerve to stand directly
under an object that's accelerating towards the ground at almost ten
meters per second squared! o_O
Also: You cannot practise high throws indoors. It doesn't work.
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