Dice Probabilities

[Kaleun]

Guys I have calculated the dice probabilities of making charges and casting throws with different rerolls.

Can you help me fill the missing gaps? Then we have it complete.

[BaldoBeardo]

Uh... why can't you finish it yourself?

[Freejack02]

Great idea for a table! The regular probabilities are pretty easy (but I'm still amazed at the amount of people that just don't bother doing it), but rerolls can really muck up the math. 

I'm curious to know how you achieved the reroll 2 results - was there a rule followed to always keep above a certain number, and how did that number change based on the target? For instance if I'm shooting for that wonderful 9" charge and I roll a 4 and a 3, obviously I should reroll the 3, but what about the 4? Was it "only reroll 1-3, because anything above 4 is above average"?

[BaldoBeardo]

1 minute ago, Freejack02 said:

Great idea for a table! The regular probabilities are pretty easy (but I'm still amazed at the amount of people that just don't bother doing it), but rerolls can really muck up the math. 

I'm curious to know how you achieved the reroll 2 results - was there a rule followed to always keep above a certain number, and how did that number change based on the target? For instance if I'm shooting for that wonderful 9" charge and I roll a 4 and a 3, obviously I should reroll the 3, but what about the 4?

Logic would always be to reroll lowest number, with higher number as 2nd. Pretty easy to brute force the figures too, with a macro and a 5x1,296 spreadsheet... 

[Freejack02]

Just now, BaldoBeardo said:

Logic would always be to reroll lowest number, with higher number as 2nd. Pretty easy to brute force the figures too, with a macro and a 5x1,296 spreadsheet... 

Yeah you are correct (and it's really just the reroll 2 I'm asking about), I'm just wondering what specifically Kaleun used to achieve those numbers. It's a bit odd that the rr2 is done, but rr1 needs to be filled out - when the math should be much more direct for that one. 

[CrimsonKing]

On the table with one reroll:

8 : 68.05555556

9 : 52.31481481

10 : 35.64814815

11: 19.90740741

12: 07.407407407

 

I cant imagine the odds of ‘one or both’, that would be tables upon tables upon tables...

[Freejack02]

4 hours ago, CrimsonKing said:

9 : 52.31481481

10 : 35.64814815

That's a larger difference than I figured it would be!

[CrimsonKing]

Also I figured out how to do one or both dice (where player has option to either reroll one or both of their choosing)

for a given set of two dice [d1,d2] compare the odds of rerolling one or both towards getting desired number, only use highest.

 [2,1]=9, the probability of one dice reroll is 0 (highest you can get is [2,6])  take the prob. for rerolling both

In some instances, typically when higher dice is 4 or better [2,4]=11, only rerolling the lowest dice produces the higher probability.

This means when you fail to roll 5 or less, better off rerolling both, anything higher and it depends on the dice.

 

[Kaleun]

Here is the updated table:

[Skarloc]

May be  you want to have a look at this AoS Combat Calculator: http://tools.druchii.net/AoS-Combat-Calculator.php

[Freejack02]

3 hours ago, Kaleun said:

Here is the updated table:

Hmmm, something is amuck here - how are the 8"+ odds better for a single reroll than a double?

[Kaleun]

4 hours ago, Skarloc said:

May be  you want to have a look at this AoS Combat Calculator: http://tools.druchii.net/AoS-Combat-Calculator.php

I find only the one die probabilities with the calc.

Quote

Hmmm, something is amuck here - how are the 8"+ odds better for a single reroll than a double?

because you may choose to reroll the smaller number if the dice. @CrimsonKing maybe you can post the formular.

[Freejack02]

33 minutes ago, Kaleun said:

because you may choose to reroll the smaller number if the dice. @CrimsonKing maybe you can post the formular.

I assumed that one/both rerolls were optional - is that not the case? Is this a table for options which force you to reroll successes? 

So "reroll 2" should be mechanically superior to "reroll 1" in every way - gaining an additional way to turn a bad dice into a better result. 

[Kaleun]

this is the problems I encountered with furfilling my table. Making 8" rolls or more with one reroll relies on choosing the smaller number... It would be different if you would reroll the higher number.

[CrimsonKing]

I assume as we are talking about AoS/40k its the different rules that say “the player can reroll one or both dice” or “the player can reroll failed charges”

isnt that the point? So when it comes down to whether you should roll one or both when two numbers are up, depends on how bad the higher dice is. One of the rules doesn't give the option.

Ill have to go another 9 hours till im home and can post but for one set when making a 7, a single reroll is still a fail; [1,1] while in most other fail  cases [1,2][1,3][2,3][3,3][2,1][2,2][3,1][3,2] you would be better off rolling both dice again. That is 9/15? Initial fail cases where reroling one (player choice, so lower number) is statistically worse, while the odds of rerolling both and getting the number (7) is the same odds as rolling in the first place, so for 3-7 is a great chance. 7 = 21/36 sets

higher targets like 9,10,11, produce lots of fail sets that are better served by keeping a single dice and rerolling one. Eg. Trying to roll 10 and rolling [1-3,6],[1-4,5] etc.

then, although the number of sets that autofail is greater, [1,1-3] for example, you have a lot of sets that have a great chance to pass when rerolling a single dice, instead of both, where the odds get worse:(6/36) chances on two dice to roll 10.

Because im bored at my lunch/communte I worked out on spreadsheet probability of rolling 3 and pick the two highest, and roll 3 and pick the 2 lowest...

3.   99.537    92.593

4.   98.148    80.093

5.   94.907    64.352

6.   89.352    47.685

7.   80.556    31.944

8.   68.056    19.444

9.   52.315    10.648

10.  35.648   5.093

11.  19.907   1.852

12.  7.407      0.463

[CrimsonKing]

My Table:

Also if it works the spreadsheet I made to calculate the above. just change the orange number to desired dice result. check 2 tabs

dice.ods

[Freejack02]

4 hours ago, CrimsonKing said:

My Table:

Also if it works the spreadsheet I made to calculate the above. just change the orange number to desired dice result. check 2 tabs

dice.ods

That table has the same result - the RR1 results are outperforming the RR2 results after a target roll of 8... which can't be right. 

Edit: Wait, is the "RR1/2" the column that represents the operator's choice of rerolling one or both? While "RR2" must reroll both? If so, then that makes sense - I didn't notice that at first. 

[CrimsonKing]

Yep 1/2 is player choice, one or both

You will notice as well that 3D6 pick highest 2 is identical to reroll 1 its the same logically as “only look at 2 of the dice at first, if those arent enough consider the 3rd dice instead”

 

Also 1 reroll outperforms 2 rerolls for high numbers because those sets where the second dice is a 5 or 6, your odds are much better keeping it than trying to roll them both. That is even true despite lots of cases where a reroll 1 autofails (if second dice was only 1-3, and reroll 1 could never reach 10-12)