El_T's Dice Probability (Updated 11/11/2013)

[El T]

My searches on this topic didn't come up with anything of the like, so here goes. Hopefully you find this helpful in experiments, theoryscape, and just general knowledge of the game.

Attack Dice
Defense Dice
More or Less

Queglix Gun SA:

• Single Hero
• Squad Figures

Also note that I will be adding my experiments that I have conducted/will conduct to this thread when possible.

[wriggz]

Okay, so you are showing a fairly complicated way to calculate the odds of rolling a given number of skills.

I'm unclear what the practical benefit is here or what you are trying to prove. Would love to see more however.

Boone Chance.

[El T]

I was in the process of doing some calculations for an experiment I'm currently conducting (I'll link you to it here when it's complete) and I had to calculate probabilities of outcomes of dice rolls. I was changing the variables of defense dice rolled, and separately, attack dice rolled. I used tree charts to determine fractional values for the probability of rolling a certain amount of skulls depending on the amount of dice rolled, and the same for defense dice. This was all fine and dandy until I got up to four dice. That took a LOT of paper to figure that one out. Five was nearly impossible. But I was really interested in this, and determined to prove my thesis. So at some point, I pulled out my handy dandy ol' TI-Nspire and plugged my current data into some tables. I began to notice some patterns...

For instance, notice how the probability of rolling zero skulls is always the same as rolling all skulls? And the probabilities increase as you approach the middle number of skulls? This is because on a die, there are 3 skulls out of 6 sides, so the ratio is 1/2. The median, as I refer to it, is always at the 'middle' amount of skulls rolled. (e.g. 2 skulls out of 4 dice or 1 skull out of 2 dice)

In my table, you'll notice, I refer to dice rolled as the term dn, where n is the amount of dice rolled. This is just because I couldn't use subscript in the table, so it really should look like Dn. (I notice I also can't use subscript in the forums, so the smaller font will have to do) In formulas, I will use the same concept for skulls rolled, as Sn.

Regardless, a formula for 0 skulls rolled can be expressed as such: (d being the independent variable of the amount of dice rolled)

To take advantage of the pattern of S0=Sd, (Sd being the probability of rolling the same amount of skulls as dice) one can use the very same formula as above to find the probability of rolling all skulls.

Now, to figure out S1 and Sd-1, it gets a tad trickier. Since the relationship between S0 and S1 varies as the amount of dice rolled varies, as shown in the table above, I knew I had to multiply the formula for S0 by a factor containing a variable. Since in D0 S0=1 and S1=0; and in D1 S0=1/2 and S1=1/2; and in D2 S0=1/4 and S1=1/2; you notice that S0 is always being multiplied by the value of d.
Therefore, the formula for S1 and Sd-1 is as follows:


After that, a pattern begins to emerge for skull probability as the amount of skulls one is solving for increases. Look at the probabilities for the next two skull values:


Therefore, the 'formula' for devising formulas for probability in attack dice is:


This may be a bit confusing, I may not be so great at explaining this all efficiently. I'll show you the fractional probabilities of all skull values, for dice 0 through 10.


For practical purposes, I didn't find it necessary to calculate beyond ten dice. If it's really necessary to determine the probabilities for more than ten dice, well that's why the "formula for formulas" that I explained earlier to come in handy.

[El T]

Originally Posted by wriggz

I'm unclear what the practical benefit is here or what you are trying to prove.

Fair point. The reason I came up with these was to aid in an experiment I was doing that involves calculating odds of particular outcomes of particular scenarios, and that involves calculating dice roll probabilities. I figure that if other people do things like that, might come in handy.

[Sir Yeshua]

Not to rain on your parade or anything, but is this what you're doing?

[El T]

Originally Posted by Sir Yeshua

Not to rain on your parade or anything, but is this what you're doing?

Meh. I suppose that's pretty similar.
To my credit, though, I devised and explained a formula for it...

[wriggz]

Originally Posted by El_T-Steff

Meh. I suppose that's pretty similar.
To my credit, though, I devised and explained a formula for it...

The formula is at the bottom of the posts, however I think your explanation is excellent. I really appreciate the work you did, Looking forward to seeing the Same treatment of the defense dice.

- to be added to the library.

[Ninja Status]

Originally Posted by wriggz

The formula is at the bottom of the posts, however I think your explanation is excellent. I really appreciate the work you did, Looking forward to seeing the Same treatment of the defense dice.

- to be added to the library.

That thread is pretty similar to your thread El_T but it is always good to have a different thread that is a little different from the others.

[El T]

Originally Posted by wriggz

The formula is at the bottom of the posts, however I think your explanation is excellent. I really appreciate the work you did, Looking forward to seeing the Same treatment of the defense dice.

- to be added to the library.

Hmm. I guess i probably should have read that article all the way through before defending myself, eh?

But anyway thanks wriggz, I figure I have no reason to stop putting my data up.
I can't right now, because most of my browsing on the site is from my phone (as i am right now) and I can only really use the screenshots and such from my computer.

[El T]

Originally Posted by Ninja Status

That thread is pretty similar to your thread El_T but it is always good to have a different thread that is a little different from the others.

Thanks NS, I think you're right.

[El T]

The formula for defense dice is very similar in some ways to the formula for attack, it's just a bit more complicated because the median (which i refer to in my Attack Dice article)

Originally Posted by El_T-Steff

For instance, notice how the probability of rolling zero skulls is always the same as rolling all skulls? And the probabilities increase as you approach the middle number of skulls? This is because on a die, there are 3 skulls out of 6 sides, so the ratio is 1/2. The median, as I refer to it, is always at the 'middle' amount of skulls rolled. (e.g. 2 skulls out of 4 dice or 1 skull out of 2 dice)

... is 1/3 of the way through the data, unlike attack dice in which the median is dead center. This is because, as you all know, there are two shields on a die out of 6 sides. therefore the ratio of shields to blanks is 1/3.

My process for deciphering the defense dice probability formula basically goes along the same lines as my attack dice formula, so I won't bore you with the details.
I will, however, explain in great detail my findings.


Since the defense dice median is off-center, for lack of a better term, S1 does not equal Sd-1. There still are formulas for Sd-n that I figured out, but it isn't nearly as helpful to use them for defense dice.

Anyways, here are the formulas for the first 4 shield values:

Notice the pattern?

The 'formula for the formula' is:

As I said, similar to attack dice.

The full set of probabilities for dice 0 through 10 is this:

[orgsbane]

Good stuff Wish I was good at math like this