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====== Dominions Random Number (DRN) ======

Most Dominions game mechanisms use something called the **Dominions Random Number (DRN**). This includes virtually all Damage rolls, [[attack skill|Attack]] rolls, [[Precision|Missile Hit]] rolls, and so on. When a random number is called for, the number used is usually actually a DRN. This is a roll of two six-sided dice (2d6) but with an additional bonus: if any individual die roll is “6,” one is subtracted, and then that die is re-rolled and added to the result. This is referred to as an “open-ended” 2d6 roll. The DRN is two six-sided dice. A lower-case "drn" is one six-sided die.

Example: The game calls for a DRN. Two dice are rolled and they come up 2,6. Because one of the
dice was a “6,” one is subtracted from the total (making 7), and the die is rolled again. But this die is
also a 6. So one is subtracted from the total (now up to 12) and a die is rolled again. It is a 4. The final
result for this DRN is 16.

Note that if both original dice came up as 6, both would be re-rolled and added as above. If a die keeps coming up 6, it keeps getting re-rolled and added, which can very occasionally lead to large numbers. Dominions has a lot of situations where something is very unlikely to happen, like a militia soldier hitting an ethereal monster. However, in the real world of Dominions, very few things are actually impossible. To model this fact as closely as can be, the Dominions Random Number was created. With it, it is always possible for such an event to occur, which would not be the case if the roll was not openended. In some very few cases, there may be only one six-sided die rolled. It is still open-ended, but in this case, the rules refer to it as a drn, in lower-case letters. An example is the dispelling of global enchantments. 


====== Probabilities in Dominions 5 ======

Most die rolls in Dominions 5 involve one player rolling higher than another player using the DRN system. To give players some idea of how likely something is to happen, here is a table that shows the difference between two values on the left, and the chance of beating that value using two open-ended dice on the right.

<WRAP group>
<WRAP column quarter>
^ Difference ^ Chance ^ 
| -30 | 0.006% | 
| -29 | 0.008% | 
| -28 | 0.012% | 
| -27 | 0.017% | 
| -26 | 0.023% | 
| -25 | 0.031% | 
| -24 | 0.043% | 
| -23 | 0.060% | 
| -22 | 0.082% | 
| -21 | 0.11% | 
| -20 | 0.16% | 
| -19 | 0.22% | 
| -18 | 0.30% | 
| -17 | 0.41% | 
| -16 | 0.55% | 
| -15 | 0.76% | 
| -14 | 1.0% |
| -13 | 1.4% |
| -12 | 1.9% |
| -11 | 2.6% |
</WRAP>
<WRAP column quarter>
^ Difference ^ Chance ^ 
| -10 | 3.4% |
| -9 | 4.6% |
| -8 | 6.2% |
| -7 | 8.2% |
| -6 | 11% |
| -5 | 14% |
| -4 | 18% |
| -3 | 24% |
| -2 | 30% |
| -1 | 38% |
| **0** | **46%** |
| 1 | 54% |
| 2 | 62% |
| 3 | 70% |
| 4 | 76% |
| 5 | 82% |
| 6 | 86% |
| 7 | 89% |
| 8 | 92% |
| 9 | 94% |
</WRAP>
<WRAP column quarter>
^ Difference ^ Chance ^ 
| 10 | 95% |
| 11 | 97% |
| 12 | 97% |
| 13 | 98% |
| 14 | 98.6% |
| 15 | 99.0% |
| 16 | 99.2% |
| 17 | 99.4% | 
| 18 | 99.6% |
| 19 | 99.7% |
| 20 | 99.79% | 
| 21 | 99.84% |
| 22 | 99.89% |
| 23 | 99.92% |
| 24 | 99.94% |
| 25 | 99.96% |
| 26 | 99.97% |
| 27 | 99.978% |
| 28 | 99.988% |
| 29 | 99.991% |
</WRAP>
</WRAP>
What does this mean? It means that if you have a Jotun Moose Rider with attack skill 9 and your opponent has an Abysian Infantry with defense skill 10, your chance of beating him with two openended dice (and thus scoring a hit) is 38%. If the values were reversed, your chance of success would be 54%. Why the seeming disjunction? Because the “zero-point” is only 46%. Remember – the table shows the chance of beating your opponent. Thus, if you are evenly matched, you need to roll higher than he or she does on the same type of dice, and thus your chances of doing so are less than even. 46%, to be exact. 

Sometimes the manual will state that a given effect requires a morale check (or some other ability check) “against” some number. This is simply a way of saying that a unit’s morale (or other ability) + DRN is compared to the stated number + DRN. So if a unit has to “take a morale check against 12,” this means the unit’s morale + a DRN is compared to 12 + DRN. If the unit has a morale of 10, the chart above would indicate that the chance of this unit passing the check is 30%.

===== Rolls for success =====
A common situation you'll find yourself in is trying to win one at least one of a series of particularly difficult rolls, such as trying to get a [[Soul Slay]] cast through the [[Magic Resistance]] of a [[super combatant]]. Here are a few values to orientate yourself around what to expect: 

^ Difference  ^ Single roll success  ^ Expected rolls for success (1 in ...)  ^ Rolls needed for >80% chance of one success  ^ Examples for magic pen against [[Magic Resistance]] of 18             ^
| 0           | 46%                  | 2                                      | 3                                            |                                                   |
| -1          | 38%                  | 3                                      | 4                                            |                                                   |
| -2          | 30%                  | 3                                      | 5                                            |                                                   |
| -3          | 24%                  | 4                                      | 6                                            | Hard to resist magic pen roll                     |
| -4          | 18%                  | 6                                      | 9                                            |                                                   |
| -5          | 14%                  | 7                                      | 11                                           |                                                   |
| -6          | 11%                  | 9                                      | 14                                           |                                                   |
| -7          | 8.20%                | 12                                     | 19                                           | Standard magic pen roll                           |
| -8          | 6.20%                | 16                                     | 26                                           |                                                   |
| -9          | 4.60%                | 22                                     | 35                                           |                                                   |
| -10         | 3.40%                | 29                                     | 47                                           |                                                   |
| -11         | 2.60%                | 38                                     | 62                                           | Easily negated magic pen roll                     |
| -12         | 1.90%                | 53                                     | 84                                           |                                                   |
| -13         | 1.40%                | 71                                     | 115                                          |                                                   |
| -14         | 1.00%                | 100                                    | 161                                          |                                                   |
| -15         | 0.76%                | 132                                    | 211                                          | Easily negated + ??Antimagic?? source  |
| -16         | 0.55%                | 182                                    | 292                                          |                                                   |
| -17         | 0.41%                | 244                                    | 392                                          |                                                   |
| -18         | 0.30%                | 333                                    | 536                                          |                                                   |
| -19         | 0.22%                | 455                                    | 731                                          |                                                   |
| -20         | 0.16%                | 625                                    | 1006                                         |                                                   |
| -21         | 0.11%                | 909                                    | 1463                                         |                                                   |
| -22         | 0.08%                | 1220                                   | 1962                                         |                                                   |
| -23         | 0.06%                | 1667                                   | 2682                                         |                                                   |
| -24         | 0.04%                | 2326                                   | 3743                                         |                                                   |
| -25         | 0.03%                | 3226                                   | 5191                                         |                                                   |
| -26         | 0.02%                | 4348                                   | 6997                                         |                                                   |


===== Understanding the DRN =====

Due to how the DRN works, there are few skill checks that are utter guarantees, and determining what the result of a roll will be is even more difficult.

==== drn Math (One Die) ====

You could zoom in on the odds of an open-ended six-sided die's results an infinite number of times, but let's just say you only expand twice at most:

<WRAP group>
<WRAP half column>
**Out of 216 Closed Rolls...**
  * 36 will be 1's (16.67%)
  * 36 will be 2's (16.67%)
  * 36 will be 3's (16.67%)
  * 36 will be 4's (16.67%)
  * 36 will be 5's (16.67%)
  * 36 will be 6's (16.67%)
**Average: 3.5**

</WRAP>

<WRAP half column>
**Out of 216 Open Rolls...**
  * 36 will be 1's (16.67%)
  * 36 will be 2's (16.67%)
  * 36 will be 3's (16.67%)
  * 36 will be 4's (16.67%)
  * 36 will be 5's (16.67%)
  * 6 will be 6's (2.778%)
  * 6 will be 7's (2.778%)
  * 6 will be 8's (2.778%)
  * 6 will be 9's (2.778%)
  * 6 will be 10's (2.778%)
  * 1 will be an 11 (0.463%)
  * 1 will be a 12 (0.463%)
  * 1 will be a 13 (0.463%)
  * 1 will be a 14 (0.463%)
  * 1 will be a 15 (0.463%)
  * 1 will be at least 16 (0.463%)
**Average: 3.930** (if the 16+ roll is ignored; **~4** otherwise)

</WRAP>
</WRAP>

==== DRN Math (Two Dice) ====

<WRAP group>
<WRAP half column>
**Out of 1296 Closed Rolls...**
  * 36 will be 2's (2.778%)
  * 72 will be 3's (5.556%)
  * 108 will be 4's (8.333%)
  * 144 will be 5's (11.11%)
  * 180 will be 6's (13.89%)
  * 216 will be 7's (16.67%)
  * 180 will be 8's (13.89%)
  * 144 will be 9's (11.11%)
  * 108 will be 10's (8.333%)
  * 72 will be 11's (5.556%)
  * 36 will be 12's (2.778%)

**Average: 7**
</WRAP>

<WRAP half column>
**Out of 1296 Open Rolls...**
  * 36 will be 2's (2.778%)
  * 72 will be 3's (5.556%)
  * 108 will be 4's (8.333%)
  * 144 will be 5's (11.11%)
  * 180 will be 6's (13.89%)
  * 156 will be 7's (12.04%)
  * 132 will be 8's (10.19%)
  * 108 will be 9's (8.333%)
  * 84 will be 10's (6.481%)
  * 60 will be 11's (4.63%)
  * 51 will be 12's (3.935%)
  * 42 will be 13's (3.241%)
  * 33 will be 14's (2.546%)
  * 24 will be 15's (1.852%)
  * 15 will be 16's (1.157%)
  * 12 will be 17's (0.926%)
  * 8 will be 18's (0.617%)
  * 7 will be 19's (0.54%)
  * 3 will be 20's (0.231%)
  * 4 will be 16's with another roll (0.309%)
  * 4 will be 17's with another roll (0.309%)
  * 4 will be 18's with another roll (0.309%)
  * 4 will be 19's with another roll (0.309%)
  * 4 will be 20's with another roll (0.309%)
  * 1 will be a 20 with two more rolls (0.072%)
**Average: ~8** (assuming the extra rolls add either drn average)

</WRAP>
</WRAP>

===== Dueling DRNs =====

//Most// DRN rolls are in opposition to another DRN roll. The Damage roll, for example, is opposed by a [[Protection]] DRN roll. The Attack roll is opposed by a [[Defence Skill|Defence]] roll, the Penetration roll to make many [[magic]] spells work is opposed by a [[Magic Resistance]] roll, and so on. In these three cases, a base value (typically matching the name of the roll) is added on to whatever the result is, and what happens next is determined by the value.

  * A hit is considered to hit if the Attack roll **beats** (doesn't tie with) the Defence roll, at least without any Parry bonus.
  * A hit is considered to **do damage** if the Damage roll **beats** the Protection roll. The difference determines the damage done, but some attacks have "side effects" (such as the [[poison damage|poison]] of an ??Asp?? bite) that only check if the hit did damage at all.
  * A Penetration roll succeeds if it **at least ties with** the opposing Magic Resistance roll.
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