Applications of Math in Guardians, Part 3: Dice Rolls

In our previous discussion of Math in Guardians, we looked at the subject of probability with regard to Planes of Entropy. This time we're going to expand that discussion to include all cards that use dice rolls in Guardians.

The Drifter's Nexus expansion introduced dice rolling into Guardians, largely due to the introduction of Entropy, also known as Chaos, and even sometimes called randomness. This was merely an experiment, it would seem, because the next expansion, Necropolis Park, doesn't have a single card that requires a dice roll. I revisited the concept with a few cards in my Champion's Odyssey expansion (including my Guardian), but to this day only 15 cards out of a total of 675 official cards use dice rolls. Those 15 cards are:

Demorgan the Inciter
Disc of Siin
Drifter's Nexus Stronghold Center
Drifter's Nexus Stronghold Left
Drifter's Nexus Stronghold Right
Garuda Kahn, First Disciple
Initiate of Entropy
Mendu Sada, the Havoc
Mighty Tiki God
Orella of the Mist
Planes of Entropy
Professor Heisenburg
Summon Entropy Storm
Vikia Tso'Shan'Lu
Xaz, Thief of Twilight

In a moment we'll take a closer look at these cards and how they are affected by probability. First, let's dive into some math that I mentioned in the last installment: Distribution and approaching the mean.

DISTRIBUTION & APPROACHING THE MEAN

Remember that I stated that averages, when used for probability, do indicate that we will flip an equal number of heads and tails over time. This is called a normal distribution. This also applies to dice rolls. To start, let's define what Mean is:

Arithmetic Mean is simply the sum of a group divided by the number of samples in the group. For instance, if you rolled a D6 6 times, getting a unique result each time, you would have results of 1, 2, 3, 4, 5, 6. Add those numbers together (21) and divide by the number of times you rolled (6). You get an average of 3.5. This 3.5 is our mean, an averaging out of unique rolls in our range.

Normally you don't want to use mean to calculate averages, because very high or low numbers can skew the data. If I live in the same town as Bill Gates, my income level & property values would be below the mean, because the income & property value of Bill Gates raises the mean greatly, so median is used instead (median income, median property value). But for rolling dice using mean is perfectly acceptable. There is a fixed range, 1 through 6, with an equal chance to roll any number.

As we continue to roll dice, our results will shift around but will eventually approach the mean.

To provide an example, I rolled a d6 100 times. These are the number of times I rolled each number:

1 = 13, total value = 13 (1 x 13)
2 = 18, total value = 36 (2 x 18)
3 = 20, total value = 60 (3 x 20)
4 = 16, total value = 64 (4 x 16)
5 = 23, total value = 115 (5 x 23)
6 = 10, total value = 60 (6 x 10)

Adding up the total value results in 348. Dividing 348 by 100 gives a mean of 3.48, which rounded off is 3.5, the same as what we established above.

Ok, you're saying, how is this useful? Think of it this way: for every 1 you roll, you'll roll a 6 (1 + 6 = 7, and 7/2 = 3.5). For every 2 you roll, you'll roll a 5 (2 + 5 = 7, and 7/2 = 3.5). For every 3 you roll, you'll roll a 4 (3 + 4 = 7, and 7/2 = 3.5). While that doesn't translate out to my 100 rolls exactly, it's pretty close. For example, I rolled 13 1's and 10 6's, only a difference of 3 per 100 rolls. Put into perspective for cards with a D6 roll, most have poor results with a low roll and optimal results with a high roll. Therefore, even if you roll high and get good results, approaching the mean indicates that you will eventually get corresponding low rolls that will hamper your strategies.

RUNS

Runs are results that consistent and concurrent, such as flipping 5 heads in a row. In my 100 D6 rolls, I never rolled more than 2 of the same number in a row. Looking at a grouping of numbers, such as rolling a 5 or a 6, I had one run of 6-5-5. Conversely, I had a runs of 1-2-1-2-2 and 1-1-2-1. Runs (which you might also call hot or cold streaks) make it seem like you are beating the averages, but as you can see from my totals, despite the runs you eventually end up approaching the mean.

Now, let's look at approaching the mean, combined with a revisiting of our mutually exclusive union, as they apply to the cards with dice rolls.

DRIFTER'S NEXUS STRONGHOLD

The Drifter's Nexus stronghold provides a 1D6+1 Vitality bonus. Here's what the other strongholds supply for bonuses (regardless of their other abilities):

Carreg Amroth = +5
Freebooter = +6 center, +4 left & right
Khnumian = +3
Necropolis Park = +3
Sabu Amantek = +4 center, +3 left & right

Drifter's Nexus has a range of +2 to +7. This means it can give you the best defense bonus, but it can also provide the worst. Without a special ability, the only reason to choose Drifter's Nexus is to gain the best defense bonus (at least equal to Carreg Amroth), +5, +6 or +7, which requires a roll of 4, 5 or 6. Our union probability tells us that the chance of this happening is 50%. Conversely, you have a 50% chance of rolling a 1, 2 or 3. This makes the Drifter's Nexus an unreliable defensive Stronghold. Half the games you play would have a good bonus, and the other half would not provide a good bonus. And in the context of approaching the mean, for each time you gain that +7, you're eventually going to get hit with a +2. Thinking back to my run of 1-2-1-2-2, if these were Drifter's Nexus rolls, I would have had a string of games with +2, +3, +2, +3, and +3, during which I'd be tearing my hair out in disbelief at my horrible dice rolling.

DISC OF SIIN AND MIGHTY TIKI GOD

Disc of Siin prevents channeling against your first # of primary attackers based on a D6 roll: 1 = no effect, 2 = 1, 3 = 2, 4 = 2, 5 = 3, 6 = all. Given that most Shields have 4-5 creatures under them, you have a 67% chance of protecting only your first 2 creatures. By contrast, an Ancient Ogre shuts down all opponent channeling for the remainder of combat. Disk of Siin has a few advantages over Ancient Ogre, however. First, it doesn't take up any Vitality under your Shield, while an Ancient Ogre takes 13 Vitality. Second, the Disc starts working on your first primary attacker, while Ogre doesn't start working until put into play, leaving some creatures unprotected if Ogre is not your first attacker. Third, you roll before primary attacks begin; therefore if you have a creature you want to protect from channeling, you know which spot to play it in to protect it. With an 84% chance to protect at least one creature, all things considered Disc of Siin is not a bad card.

Mighty Tiki God allows you to add 1D6 of Vitality to a primary matchup creature for each bribery card discarded, giving you a range of +1 to +6 per bribery card. You are just as likely to get a +1 as you are a +6, but ultimately you will approach the mean (3.5). How does this compare with other cards that give you combat bonuses? Most bonuses come with strings attached, such as working in a specific terrain or against large creatures. In fact, I can't find any card that gives all your creatures an unconditional bonus, except maybe Grand Poobah Schnee. In the right conditions (say, against an Undead deck with no bribable creatures), tossing bribery cards makes Mighty Tiki God very useful.

PROFESSOR HEISENBURG AND SUMMON ENTROPY STORM

Professor Heisenburg is an amazing card when it works. However, for your range attacker to destroy the opponent, you have to roll a 5 or 6, which would only happen 1 out of every 3 rolls. That means that 66% of the time you use it, the Professor is a wasted card.

Summon Entropy Storm, like the Professor, is a wasted card because 66% of the time you destroy your own creature. The only saving grace of the card is the fact that small creatures get +1 to the roll. Therefore a Tookle deck may want to consider it, because this changes the mean from 3.5 to 4.5. That's huge, because now a roll of 4-6 (50% chance) is successful. When you have fairies taking down Devils, Knights, Ogres, Dragons, and other bigger creatures every other time you play Summon Entropy Storm, it seems like a no-brainer.

DISCIPLES OF ENTROPY

Initiate of Entropy gets 1D6 added to its vitality. It has a Base Vitality of 2 and a stacking penalty of 5. You must roll a 3 for the vitality to equal the stacking penalty, and 4-6 to exceed it. Not a bad deal, since you have a 66% chance of rolling 3 or higher. Add in Mu Kir' Agavati's ability to re-roll and Initiate looks pretty good, especially if you roll a 5 or 6 - that's a Vitality of 7-8 that only stacks as 5! Plus you can channel 2 more if needed.

Demorgan the Inciter is truly unique - its the only card in the game that uses 2D6. It has a Base Vitality of 6 with a stacking penalty of 11. We determined that the mean for 1D6 is 3.5; for 2D6, the mean is 7. Since this is what you will average over time, that means Demorgan will average a 13 Vitality and only stack as 11. Looking at it another way, there are 11 equal chances to roll 2 through 12. You have a 9% chance to roll any number, and you must roll a 5 to equal the stacking penalty. You have a 73% chance to roll 5 or better - this is outstanding! Rolling double sixes gives you 18 Vitality with only an 11 stacking penalty! And we haven't even considered Mu Kir' yet...let's say you roll a 5 on one die and a 2 on the other. You can use Mu Kir' to re-roll that 2, knowing you're guaranteed at minimum a Vitality bonus of 6. One of the best cards in the game, period.

Mendu Sada is the not the worst of cards, but his ability is almost useless. An AOE of 1 or 2 is lousy, and rolling a 6 has no effect. That means half the time his ability is worthless. Add to the fact that he stacks as 11, when you might already have Demorgan or Garuda Kahn stacking as an 11, and there's really no place for Mendu except maybe against Tookle decks.

Orella of the Mist is also somewhat useless, as she only has a 33% chance to destroy the creature she faces and is only a 4 Vitality stacking as 7. Remember that re-rolling the dice with Mu Kir' doesn't change that 33% chance, it only gives a you a second try at it. Combined with Dragon Standard Bearer 19, however, she becomes a killing machine for only 1 Power Stone...roll your D6 first, then change the border color to match the die roll and watch her take down a Watcher or Eternal Witch Lord easily. She can even receive channeling!

Vikia Tso'Shan'Lu and Xaz, Thief of Twilight are very similar. Their abilities trigger on a roll of 3 or better, which you have a 66% chance of doing. Vikia stacks better as a 4, but must beat her opponent to trigger the ability, making it much harder to use, although she can accept channeling. Xaz just has to be in play to trigger the ability, but does take up a little more room with a 6 Vitality.

Garuda Kahn is last but not least. This guy was built to kill Guardians, and pretty much everything else in the game. It's as close to a "broken" card as you will find. Garuda is a 9 that stacks as 11. You want to play other Initiates with him to make use of his ability. That means you probably already have a Mu Kir' in play to re-roll any dice, and possibly an Initiate. That's going to give Garuda a 2D6 bonus, same as Demorgan, but starting at 9 Vitality instead of 6. Save him as a secondary attacker and you can get up to 4D6 or 5D6, allowing him to crush anything, including Guardians. This guy's only weaknesses are Beer and any effect that removes your Disciples from play.

That concludes part 3. There may be a part 4 in the future, but for now I'm moving on to other projects. I hope you've gotten some useful information out of the series.

p.s. I'll put some card scans into this post soon...