Author Topic: Huh. This looks interesting...  (Read 4310 times)

Offline Amechra

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Huh. This looks interesting...
« on: September 02, 2013, 08:19:23 PM »
Alright, I was messing around with probabilities a bit, and I found a really interesting "set" of probabilities.

Namely, what happens when you roll 1dX + 1d(X + 4).

I'll give an example: 1d8 + 1d12 vs. 2d10.

The average values are the same; however, once you graph out the probabilities, you find that there is an equal chance to roll any value from 9 to 13 on the 1d8 + 1d12.
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Offline Garryl

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Re: Huh. This looks interesting...
« Reply #1 on: September 02, 2013, 09:08:33 PM »
For any 1dX + 1d(X+Y), where Y>=0, the middle (Y+1) values will have equal probabilities.

Edit: Whoops, math fail.
« Last Edit: September 02, 2013, 09:22:00 PM by Garryl »

Offline Raineh Daze

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Re: Huh. This looks interesting...
« Reply #2 on: September 02, 2013, 09:14:49 PM »
For any 1dX + 1d(X+Y), where Y>=0, the middle (Y-X+1) values will have equal probabilities.


Sure you've written that right? Because for 1d8 +1d12, that's 1d8 + 1d(8+4), which leads to the middle 4-8+1 = -3 values. When it's a range of 4.

Now, what I want to know is how it's skewed like that, and EQUALLY...

Offline Amechra

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Re: Huh. This looks interesting...
« Reply #3 on: September 02, 2013, 09:32:51 PM »
Should be (X - Y + 1).

It reminds me of the old "saw" that rolls are most exciting when you've got between a 25% and a 75 % chance of success.

I mean, this theoretically combines the better parts of a d20 and a bell curve...
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Offline Raineh Daze

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Re: Huh. This looks interesting...
« Reply #4 on: September 02, 2013, 09:34:08 PM »
Should be (X - Y + 1).

It reminds me of the old "saw" that rolls are most exciting when you've got between a 25% and a 75 % chance of success.

I mean, this theoretically combines the better parts of a d20 and a bell curve...

... not seeing it.

Offline Amechra

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Re: Huh. This looks interesting...
« Reply #5 on: September 02, 2013, 09:39:07 PM »
Then it might just be me being crazy.
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Offline Garryl

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Re: Huh. This looks interesting...
« Reply #6 on: September 02, 2013, 09:42:53 PM »
Sorry, it's supposed to be the innermost Y+1 being the same, not Y-X+1 (which was the case when I was thinking about it as 1dX + 1dY instead). The outermost X-1 values on each side will have the standard sort of linearly-decaying distribution you normally see from rolling two dice.

As for why, consider, one die alone (specifically, the larger one). All values have an equal chance of coming up, from 1 to X+Y. Then add in another die, with equal chances of 1 through X. The larger die has Y more numbers than the smaller die that it can generate. The innermost combinations of values (in a simple example of 1d4+1d10, those would be 5 through 11) can be generated using any values on the smaller die and an appropriate value on the larger die, but the outermost values (2, 3, 4, 12, 13, and 14) can only be generated using some values on the smaller die; others do not have an appropriate value on the larger die with which to generate them. For example, 7 can be made with 1+6, 2+5, 3+4, and 4+3, by 12 can only be made with 2+10, 3+9, or 4+8; a 1 on the d4 doesn't do it. So, the innermost values all have the same number of chances of coming up (4/40, in the example above), but the outermost values have the usual linearly decaying probabilities that you would expect from rolling two dice together.

Edit: A bit more math fail. It's X-1 on each side, not X in total.
« Last Edit: September 02, 2013, 10:29:35 PM by Garryl »

Offline Raineh Daze

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Re: Huh. This looks interesting...
« Reply #7 on: September 02, 2013, 09:54:17 PM »
But why is 1d8+1d12 skewed towards higher numbers? That's the confusing bit.

Offline Amechra

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Re: Huh. This looks interesting...
« Reply #8 on: September 02, 2013, 10:03:01 PM »
It isn't. Remember, the minimum value is 2, and the average value is 11.

It just looks skewed, even though the "flat" region is centered on the average roll.
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Offline Raineh Daze

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Re: Huh. This looks interesting...
« Reply #9 on: September 02, 2013, 10:04:11 PM »
... it's late and I forgot that 10 existed. Um. Oops. So I was thinking it was 9, 11, 12, 13...

Offline Agita

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Re: Huh. This looks interesting...
« Reply #10 on: September 02, 2013, 10:26:50 PM »
Should be (X - Y + 1).

It reminds me of the old "saw" that rolls are most exciting when you've got between a 25% and a 75 % chance of success.

I mean, this theoretically combines the better parts of a d20 and a bell curve...

... not seeing it.
Theoretically, you might get an even distribution of probabilities within the 25-75% range of the roll (d20's advantage), if you go with that being the most exciting range of success, while results outside that range have a decreasing chance of happening (bell curve's advantage).
However, it's missing the other big advantage of d20 - that probabilities are extremely easy to calculate on the fly given a bonus and a DC.

That said, this is a fun find and thought experiment, but I don't think it would be practical in actual application as a core dice mechanic. Having to roll two differently-sized dice in order to get your base value is rather too clunky.
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Offline Amechra

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Re: Huh. This looks interesting...
« Reply #11 on: September 02, 2013, 10:27:55 PM »
Alternity does it just fine, though.
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Offline nijineko

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Re: Huh. This looks interesting...
« Reply #12 on: September 03, 2013, 01:05:33 PM »
Alternity does it just fine, though.

so does the Feng Shui rpg.