(TFT) Re: bell curves, etc.

[Mark Tapley <mtapley@swri.edu>]

At 3:13 PM -0500 1/27/09, Jay wrote:
> Take a cube (die) 1" by 1" by 1".
> Truncate it by marking the middle point of each edge and removing the corners.
> Six square sides and 8 triangle sides.
> So what is the curve for three 14 sided die?
> After many MANY empirical tests, a.k.a. rolls, I can state with some 
> confidence that the thing as described rolls a corner, or triangle, 
> 1 time out of 5.
> I strongly suspect the things probability to be directly related to 
> its size, due to the differences in surface areas.
> (Vaguely interesting math to that, what. Center point of the 
> trunk-cube to the center point of each surface..., I'll have to 
> spend a shcoosh more time here.)

Very interesting math. Stability of that die sitting on a triangle is 
different from that of the die sitting on a square. Although the 
probability of its coming to rest on any given triangle is the same 
as that on any other triangle, the relative probability between a 
triangle and a square depends on *lots* of factors other than 
geometry. Resiliance and stiffness of the die itself as well as the 
material it's bouncing on, sliding and sticking coefficients of 
friction between the die and the surface, angular momentum vs. linear 
momentum at impact, etc. etc. etc.

In other words, it's not a platonic solid, and in fact it's not a 
solid where each face is "identical" in any sense to the other faces, 
and so it's not a "fair" die in the sense that it doesn't randomize 
linearly (or even predictably, I'd say) between triangles and squares.

It may be an interesting study, but I would not want to game with it.

At 3:13 PM -0500 1/27/09, Neil wrote:
> Someone with a DX of 10 puts in the same amount of learning time, and
> gets a DX of 11. They get a jump from 50% to something like 67%,
> because they're actually learning more in the same amount of time.

I just realized something I'd never noticed before with respect to 
the TFT "bell curves" for multiple dice. Most of y'all are probably 
way ahead of me on this and can ignore it, but:

Although the bell curves are in fact steeper for more than 3 dice, as 
plotted on a scale normalized to the overall distribution, they are 
*shallower* than for 3 dice in terms of % increase in chance of hit 
vs. increase in adjDX.

In concrete terms, listing the *greatest* improvement in chance to hit:

on	going from DX		gives % improvement in chance to hit
------	------------------	-----------------------------
1 die	3 to 4 (any change)	16.7%
2 dice	6 to 7			16.7%
3 dice	9 - 10 or 10 - 11	12.5%
4 dice	13 to 14		11.265%
5 dice	16 - 17 or 17 - 18	10.03%
6 dice	20 to 21		9.285%
7 dice	23 - 24 or 24 - 25	8.579%
8 dice	27 to 28		8.094%

etc.

In other words, the x axis extends faster than the slope steepens, if 
you plot %chance of hit vs. number required to hit.

I have the full chart of hit probabilities on an Appleworks 
spreadsheet, and will be happy to send it along if anyone is 
interested, or describe the algorithms used.
--
						- Mark     210-379-4635
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