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 	+----------------------------------------------------TL;DR---------------------------------------------------------+
 	+When damage and protection are equal, the DRN will benefit the attacker by about 1.42 points of damage on average.+
 	+------------------------------------------------------------------------------------------------------------------+
 
-This sheet uses 2d5 rolls to model the DRN, the exploding dice have been abstracted as the average difference of a 1d5 vs. 1d5 roll against the other unit with one of the dice set at '5' (natural six implies a '5' on the previous roll), plus the average difference of an opposed 1d5 roll with your die as any number (the most recent roll), modified by the probability of rolling a natural six in the first place, or 1/6.
+
 
 This sheet uses 2d5 rolls to model the DRN, the exploding dice have been abstracted as the average damage bonus of a 1d5 vs. 1d5 roll against the other unit with one of the dice set at '5' (natural six implies a '5' on the previous roll), plus the average damage bonus of an opposed 1d5 roll with your die as any number (the most recent roll), modified by the probability of rolling a natural six in the first place, or 1/6.
 
 The odds of rolling a particular value for a 2d5 are:
 
 2: 1/25
 3: 2/25
 4: 3/25
 5: 4/25
 6: 5/25
 7: 4/25
 8: 3/25
 9: 2/25
 10: 1/25
 
 Now let's look at the possible combinations of a 2d5 vs 2d5 Protection/Damage DRN roll, assuming both stats are the same. There are (9*9) or 81 values that may result from a total of (5*5)*(5*5) or 625 unique rolls.
 
 Prot,Dam|AtkrBonus|Odds
 
 2,2=	0	1/625
 2,3=	1	2/625
 2,4=	2	3/625
 2,5=	3	4/625
 2,6=	4	5/625
 2,7=	5	4/625
 2,8=	6	3/625
 2,9=	7	2/625
 2,10=	8	1/625
 Subtotal: 25/625
 
 3,2=	0	2/625
 3,3=	0	4/625 <--note: there are two different ways to roll a '3' in a 2d5 roll,
 3,4=	1	6/625		1,2 and 2,1 so 2*2 is 4 different combinations.
 3,5=	2	8/625
 3,6=	3	10/625
 3,7=	4	8/625
 3,8=	5	6/625
 3,9=	6	4/625
 3,10=	7	2/625
 Subtotal: 50/625
 
 4,2=	0	3/625
 4,3=	0	6/625
 4,4=	0	9/625
 4,5=	1	12/625
 4,6=	2	15/625
 4,7=	3	12/625
 4,8=	4	9/625
 4,9=	5	6/625
 4,10=	6	3/625
 Subtotal: 75/625
 
 5,2=	0	4/625
 5,3=	0	8/625
 5,4=	0	12/625
 5,5=	0	16/625
 5,6=	1	20/625
 5,7=	2	16/625
 5,8=	3	12/625
 5,9=	4	8/625
 5,10=	5	4/625
 Subtotal: 100/625
 
 6,2=	0	5/625
 6,3=	0	10/625
 6,4=	0	15/625
 6,5=	0	20/625
 6,6=	0	25/625
 6,7=	1	20/625
 6,8=	2	15/625
 6,9=	3	10/625
 6,10=	4	5/625
 Subtotal: 125/625
 
 7,2=	0	4/625
 7,3=	0	8/625
 7,4=	0	12/625
 7,5=	0	16/625
 7,6=	0	20/625
 7,7=	0	16/625
 7,8=	1	12/625
 7,9=	2	8/625
 7,10=	3	4/625
 Subtotal: 100/625
 
 8,2=	0	3/625
 8,3=	0	6/625
 8,4=	0	9/625
 8,5=	0	12/625
 8,6=	0	15/625
 8,7=	0	12/625
 8,8=	0	9/625
 8,9=	1	6/625
 8,10=	2	3/625
 Subtotal: 75/625
 
 9,2=	0	2/625
 9,3=	0	4/625
 9,4=	0	6/625
 9,5=	0	8/625
 9,6=	0	10/625
 9,7=	0	8/625
 9,8=	0	6/625
 9,9=	0	4/625
 9,10=	1	2/625
 	Subtotal: 50/625
 
 10,2=	0	1/625
 10,3=	0	2/625
 10,4=	0	3/625
 10,5=	0	4/625
 10,6=	0	5/625
 10,7=	0	4/625
 10,8=	0	3/625
 10,9=	0	2/625
 10,10=	0	1/625
 	Subtotal: 25/625
 
 Total: 625/625 (as it should be)
 
 Let's look at the odds for a damage bonus on the 2d5 roll, before exploding dice are taken into consideration.
 
 Chance of +0:	355/625 56.8%
 Chance of +1:	80/625	12.8%
 Chance of +2:	68/625	10.88%
 Chance of +3:	52/625	8.32%
 Chance of +4:	35/625	5.6%
 Chance of +5:	20/625	3.2%
 Chance of +6:	10/625	1.6%
 Chance of +7:	4/625	0.64%
 Chance of +8:	1/625	0.16%
 
 Pre-Explosion 2d5 Average: 1.1328
 
 Now let's abstract the die explosions. We can represent bonus dice with a 1d5 vs 1d5 roll because there is an equal chance for either the attacker or defender getting a bonus die (it is symmetrical) and they share a probability coefficient, thus we can lump them together into one average. Muh algebra.
 
 Prot,Dam | AtkrBonus | Odds
 1,1=	0	1/25
 1,2=	1	1/25
 1,3=	2	1/25
 1,4=	3	1/25
 1,5=	4	1/25
 2,1=	0	1/25
 2,2=	0	1/25
 2,3=	1	1/25
 2,4=	2	1/25
 2,5=	3	1/25
 3,1=	0	1/25
 3,2=	0	1/25
 3,3=	0	1/25
 3,4=	1	1/25
 3,5=	2	1/25
 4,1=	0	1/25
 4,2=	0	1/25
 4,3=	0	1/25
 4,4=	0	1/25
 4,5=	1	1/25
 5,1=	0	1/25
 5,2=	0	1/25
 5,3=	0	1/25
 5,4=	0	1/25
-[Average AtkrBonus for 1d5 vs 1d5 bonus die]: 0.8
+
 
-In addition to the generating an extra 1d5 roll, a bonus die ensures the previous roll will have at least one five. This modifies the total possible outcomes and thus the average.
+Chance of +0: 	15/25	60%
 Chance of +1: 	4/25	16%
 Chance of +2: 	3/25	12%
 Chance of +3: 	2/25	8%
 Chance of +4:	1/25	4%
 
 [Average AtkrBonus for 1d5 vs 1d5 bonus die]: +0.8
 
 In addition to the generating an extra 1d5 roll, a bonus die ensures the previous roll will have at least one '5.'. This modifies the total possible outcomes and thus the average.
 
 Prot,Dam | AtkrBonus | Odds
 
 (assuming the attacker got the bonus die)
 1,5=	4	1/10
 2,5=	3	1/10
 3,5=	2	1/10
 4,5=	1	1/10
 5,5=	0	1/10
 
-[Average AtkrBonus for natural 6 vs 1d5]: 1
+
 5,1=	0	1/10
-These two data points represent the average benefit for the attacker when a bonus die is rolled, accounting for the possibility of the defender rolling a bonus die as well. The first data point represents
+
 5,3=	0	1/10
-Now that we have both of those averages, we can add them together (1 + 0.8) to get a cumulative benefit of 1.8 for the attacker in the event of a bonus die. TheseBut we need to factor in the probability coefficient for a bonus die actually occurring. Because the exploding dice are recursive, you can have a potentially infinite number of bonus dice, but the likelihood of this is infinitesimally small. I've only gone three iterations deep.
+
 5,5=	0	1/10
 
 Again, we can lump these together into one average.
 [Average AtkrBonus for natural 6 vs 1d5]: +1
-This is much harder to explain, but I will say that the subtraction done within the leftmost factors are to avoid redundantly adding to the overall bonus since the rolls are done not in parallel but serially, and that the division factors represent increasingly improbable levels of die recursion.
+
 Now that we have those two averages, we can add them together (1 + 0.8) to get a cumulative benefit of 1.8 for the attacker in the event of a bonus die. But we need to factor in the probability coefficient for bonus dice actually occurring. Because the exploding dice are recursive, you can have a potentially infinite number of bonus dice, but the likelihood of this is infinitesimally small. I've only gone three iterations deep.
 
 (1.8 - 1.1328)(2/6) + (1.8 - 0.8)(2/6/6) + (1.8 - 0.8)(2/6/6/6) = 0.287214814
     + 1.1328 (this is the average value for 2d5 vs 2d5) = 1.420014814
 
 This is much harder to explain, but I will say that the subtraction done within the leftmost factors are to avoid redundantly adding to the overall bonus since the rolls are done not in parallel but serially, I've included those for clarity. The rightmost factors represent increasingly improbable levels of die recursion.
 
 [Average Attacker Damage Bonus]: 1.42
 
 So there you have it. You can expect the DRN to benefit the attacker by about 1.42 if damage and protection are equal, and you can expect those gains to reduce with any inequality in those two stats as negative modifiers become possible.