Pellinore Superior Ride Probabilities

With the recent addition of Gold Paladin, I got around to reading Pellinore, and the card is actually amazingly strong.  Based on that information, I decided to figure out what the odds of hitting Pellinore are when calling cards from the deck.  For the sake of this example, Lop-Ear Shooter is assumed to be at 4 in the deck.

Calling Pellinore off of Vivianne:

1 Copy of Pellinore: 2.564%
2 Copies of Pellinore: 5.128%
3 Copies of Pellinore: 7.692%
4 Copies of Pellinore: 10.256%

Calling Pellinore off of Lop-Ear Shooter or Spring Breeze Messenger:

1 Copy of Pellinore: 7.692%
2 Copies of Pellinore: 14.8%
3 Copies of Pellinore: 21.873%
4 Copies of Pellinore: 28.384%

Calling Lop-Ear Shooter off of Vivianne, then calling Pellinore off of Lop-Ear Shooter:

1 Copy of Pellinore: 0.197%
2 Copies of Pellinore: 0.759%
3 Copies of Pellinore: 1.682%
4 Copies of Pellinore: 2.911%

Calling Lop-Ear Shooter off of Spring Breeze Messenger, then calling Pellinore off of Lop-Ear Shooter:

1 Copy of Pellinore: 0.592%
2 Copies of Pellinore:  2.19%
3 Copies of Pellinore:  4.784%
4 Copies of Pellinore:  8.057%

Calling Lop-Ear Shooter off of Lop-Ear Shooter off of Vivianne, then calling Pellinore off of Lop-Ear Shooter:

1 Copy of Pellinore:  .0005%
2 Copies of Pellinore: .0018%
3 Copies of Pellinore:  .0041%
4 Copies of Pellinore:  .007%

Calling Lop-Ear Shooter off of Lop-Ear Shooter off of Spring Breeze Messenger, then calling Pellinore off of Lop-Ear Shooter:

1 Copy of Pellinore:  .0014%
2 Copies of Pellinore:  .0053%
3 Copies of Pellinore:  .0116%
4 Copies of Pellinore: .0195%

I'm assuming nobody would discard for a third Lop-Ear just to attempt to hit Pellinore so further calculations will assume that you won't do this.

Total Odds of Calling Pellinore off of Vivianne:

1 Copy of Pellinore:  2.7615%
2 Copies of Pellinore:  5.8888%
3 Copies of Pellinore:  9.3781%
4 Copies of Pellinore:  13.174%

Total Odds of Calling Pellinore off of Spring Breeze Messenger:

1 Copy of Pellinore:  8.2854%
2 Copies of Pellinore:  17%
3 Copies of Pellinore:  26.6686%
4 Copies of Pellinore:  36.4605%

Are Galahad and Tsuk VP Farming ACTUALLY Good?

No.

Oh?  I have to explain why?  Sure.

Let's set some criteria for "Automated VP Farming Statistics".

Overall
Game Length (vs. non-VP Farmers)
Game Length (vs. VP Farmers)
Game Length (vs. non-attacking VP Farmers)
Winrate

The idea that Tsuk and Galahad farmers intended to instantiate was to trade Game Length vs. non-VP Farmers for Winrate.  Did that work out?  I mean, sort of.  But the payouts were marginal at best and the tradeoffs were huge.  It also had the unintended consequence of increasing the Game Length vs. VP Farmers by an even wider margin.  Some may have considered Tsuk/Galahad making it to grade 3 and having Twin-Drive!! or an 11k butt that the opponent can't hit, but this is improbable.  On average, it will take you as many turns to make it to grade 3 as the game lasts, so you won't be gaining these benefits, typically. 

 Let's put some stats down for VP Farming decks and explain how they got there.

3k Farmer (no attack)

Overall - S (Due to smaller amount of non-attack VP Farmers)
Game Length (vs. non-VP Farmers) - S+++
Game Length (vs. VP Farmers) - S
Game Length (vs. non-attacking VP Farmers) - F (longest possible games, actually)
Winrate - F (can't win unless opponent concedes)

3k Farmer (attack)

Overall - A
Game Length (vs. non-VP Farmers) - S++
Game Length (vs. VP Farmers) - A
Game Length (vs. non-attacking VP Farmers) - D (many no-attack farmers use larger bodies)
Winrate - D

4k Farmer (no attack)

Overall - S (Due to smaller amount of non-attack VP Farmers)
Game Length (vs. non-VP Farmers) - S+++
Game Length (vs. VP Farmers) - S
Game Length (vs. non-attacking VP Farmers) - F (longest possible games, actually)
Winrate - F (can't win unless opponent concedes)

4k Farmer (attack)

Overall - A
Game Length (vs. non-VP Farmers) - S++
Game Length (vs. VP Farmers) - A
Game Length (vs. non-attacking VP Farmers) - B
Winrate -D+

5k Farmer (no attack)

Overall - C- (No real reason to run 5k no-attack farming when all clans have 4k starters)
Game Length (vs. non-VP Farmers) - S++
Game Length (vs. VP Farmers) - B (More VP farmers have to trigger to hit you)
Game Length (vs. non-attacking VP Farmers) - F (longest possible games, actually)
Winrate - F (can't win unless opponent concedes)

5k Farmer (attack)

Overall - S+
Game Length (vs. non-VP Farmers) - S++
Game Length (vs. VP Farmers) - S
Game Length (vs. non-attacking VP Farmers) - A+ (only weakness is 6k non-attackers)
Winrate - D+

6k Farmer (no attack)

Overall - F (Yet this is extremely popular for some reason?!)
Game Length (vs. non-VP Farmers) - A (Harder for opponent to make rows to hit over triggers)
Game Length (vs. VP Farmers) - D (Only two decks can hit you, barring hitting a ride chain/trigger)
Game Length (vs. non-attacking VP Farmers) - F (longest possible games, actually)
Winrate - F (can't win unless opponent concedes)

6k Farmer (attack)

Overall - S
Game Length (vs. non-VP Farmers) - S++
Game Length (vs. VP Farmers) - B+ (Slow matchups against Tsuk/Galahad, which is already slow)
Game Length (vs. non-attacking VP Farmers) - S
Winrate - C-

Tsukuyomi/Galahad Chain Farmer (attack)

Overall - C
Game Length (vs. non-VP Farmers) - C+ ((Lower rating due to ride chain animations)
Game Length (vs. VP Farmers) -B- ((Lower rating due to ride chain animations and 6ks)
Game Length (vs. non-attacking VP Farmers) - B+ (Lower rating due to ride chain animations)
Winrate - C-


Tsukuyomi/Galahad Chain Farmer + Heals/PGs (attack)

Overall - D-
Game Length (vs. non-VP Farmers) - F (Lower rating due to ride chain/PG/Heal animations/game extensions)
Game Length (vs. VP Farmers) - C- ((Lower rating due to ride chain animations/PGs/Heals, and 6ks)
Game Length (vs. non-attacking VP Farmers) - B+ (Lower rating due to ride chain animations)
Winrate - C+


Tsukuyomi/Galahad Chain Farmer + Heals/PGs (non-attack)

Overall - F---
Game Length (vs. non-VP Farmers) - F---
Game Length (vs. VP Farmers) - F---
Game Length (vs. non-attacking VP Farmers) - F---
Winrate - F---
Troll Rating - S+++


So, let's take a look at the good and the bad here and explain the current trinity of VP Farming:

The three decks that seem to be the best at face value are:

5k attacking
6k attacking
Tsuk/Galahad ride chain

However, 6k and Tsuk/Galahad ride chain have a massive issue with their match-up:  Namely, The odds of riding up in Tsukuyomi off of the ride chain being less than hitting a coin flip each turn increases the duration of the games in this match-up due to the decks "taking turns" being able to hit each other without triggers.

We take for granted how amazing the Tsukuyomi ride chain is because you can ride from hand or top 5 of the deck.  However, with no selective mulligan and no hand-riding, your odds of riding up to grade 1 with Tsukuyomi/Galahad are: 43.616%.  The odds of riding grade 1 Tsuk/Galahad in two turns is 68.209%.  The odds in three turns are 82.075%.  Inverted, this gives us the odds of missing them.

Your odds of being a 5k against a 6k while running the ride chain:

Turn 1 - 56.384%
Turn 2 - 31.791%
Turn 3 - 17.925%

This means that for the first few turns, you have very likely odds as a 6k to be hitting the Tsuk/Galahad and them missing you barring triggers because they missed their ride.  Then, you flip that around when the ride does eventually happen and now you're just spending extra time having the person who was winning start to lose.  And to make matters even less consistent, triggers will throw the defensive walls being utilized into utter disarray, which could invalidate the temporary advantage that either side may have had.  Neither deck truly edges out over the other in winrate by a significant margin.  Thus, in theory, there's no reason to run the Galahad/Tsukuyomi ridechain over and against a 6k butt because that's the only match-up that both wouldn't just have an advantage in by default.

Unfortunately, because both exist and because both counteract each other (and Tsuk/Galahad also counteracts its own mirror), it means that you have a "bad matchup" for game speed on the VP Farming ladder for both decks.  This is why the 5k farmer who can hit Tsuk when Tsuk misses the chain, but will never failed to be hit by Tsuk or the 6k butt is the objectively best VP farming deck for pushing the ladder (not that VP farm decks are particularly great for this) and for speed, because it will never have a slow game.  No slow games means more chances to string VP farming opponents.  More VP farming opponents in a row means more potential to gain ranks.  Since ranks can't be lost, it's about match quantity over marginal increases to winrate that will help you climb.

Heals and PGs will improve the winrate for any of the VP farming decks (albeit by an incredibly marginal amount) AND because they prolong games, it can help you to hit your ride chain as Tsuk/Galahad, but that makes matters even worse because it's match count, not winrate, that will get you rank ups through automated farming.  The 6k deck with heals and PGs is actually just as "strong" as the Tsuk/Galahad deck and substantially faster, so it's also superior in this case.

The final tier list results are:

Best VP Farming decks - 5k Attacking (S+), 6k Attacking (S), 3k non-attacking (S), 4k non-attacking (S).

Note:  Non-attacking decks are only S tier right now because so few people are doing it.  If it gets popular and you start running into mirrors, it will drop to C tier.

Disclaimer - I am not advocating that you should or should not use an auto-clicker to VP farm.  What I am advocating for is if you're going to do it, make your experience and your opponent's experience better and do it properly.

Vanguard Zero - Probabilities

Trigger Probabilities

Single trigger (on any given drive/damage check) = 1/3  = 33.33%
Double triggers (on any given Twin-Drive!!) = 10.526%
Getting a heal trigger on any given drive/damage check (assuming 4 are ran) = 10.256%
Odds of getting at least one heal trigger when you hit two consecutive triggers =  53.846%
Odds of getting a heal trigger when you Twin-Drive!! = 19.703%
Odds of getting a heal trigger when double crit at 3 with same/more damage = 28.384%
Odds of getting a heal trigger when crit at 4 with same/more damage = 19.703%
Odds of getting a heal trigger when hit at 5 with same/more damage = 10.256%
Odds of getting two heal triggers when crit at 5 damage = 0.81%
Odds of hitting a trigger on Twin-Drive!! if you always send non-triggers to the bottom with CEO Amaterasu = 71.55%

Getting a 9-of trigger on any given drive/damage check = 23.077%
Getting an 8-of trigger on any given drive/damage check = 20.513%
Getting a 7-of trigger on any given drive/damage check = 17.949%
Getting a 6-of trigger on any given drive/damage check = 15.385%
Getting a 5-of trigger on any given drive/damage check = 12.821%
Getting a 4-of trigger on any given drive/damage check = 10.256%
Getting a 3-of trigger on any given drive/damage check = 7.692%
Getting a 2-of trigger on any given drive/damage check = 5.128%
Getting a 1-of trigger on any given drive/damage check = 2.564%
Getting a 0-of trigger on any given drive/damage check = 0%


Getting one specific trigger on a Twin-Drive!! check with 9 in deck = 41.256%
Getting one specific trigger on a Twin-Drive!! check with 8 in deck = 37.247%
Getting one specific trigger on a Twin-Drive!! check with 7 in deck = 33.063%
Getting one specific trigger on a Twin-Drive!! check with 6 in deck = 28.745%
Getting one specific trigger on a Twin-Drive!! check with 5 in deck = 24.291%
Getting one specific trigger on a Twin-Drive!! check with 4 in deck = 19.703%
Getting one specific trigger on a Twin-Drive!! check with 3 in deck = 14.978%
Getting one specific trigger on a Twin-Drive!! check with 2 in deck = 10.121%
Getting one specific trigger on a Twin-Drive!! check with 1 in deck = 5.128%
Getting one specific trigger on a Twin-Drive!! check with 0 in deck = 0%


Getting two of the same trigger on a Twin-Drive!! check with 9 in deck = 4.858%
Getting two of the same trigger on a Twin-Drive!! check with 8 in deck = 3.779%
Getting two of the same trigger on a Twin-Drive!! check with 7 in deck = 2.834%
Getting two of the same trigger on a Twin-Drive!! check with 6 in deck = 2.024%
Getting two of the same trigger on a Twin-Drive!! check with 5 in deck = 1.35%
Getting two of the same trigger on a Twin-Drive!! check with 4 in deck = 0.81%
Getting two of the same trigger on a Twin-Drive!! check with 3 in deck = 0.405%
Getting two of the same trigger on a Twin-Drive!! check with 2 in deck = 0.135%
Getting two of the same trigger on a Twin-Drive!! check with 1 in deck = 0%
Getting two of the same trigger on a Twin-Drive!! check with 0 in deck = 0%

Transraizer Probabilities

Odds of getting a grade 1 off of Transraizer = 34.211%
Odds of getting a grade 2 off of Tranraizer = 31.579%
Odds of getting a unit off of Transraizer = 65.789%
Odds of getting a Transraizer off of a Transraizer = 7.895%
Odds of getting a unit off of Transraizer off of a Transraizer = 5.121%
Odds of Transraizer whiffing = 34.211%

I was going to make a section on riding probabilities, but the odds of misriding are so low and we live in a 9 draw meta, so I can't imagine that it's all that necessary.  Especially with ride-fixing cards in the game such as Gancelot and Aleph.  Thus, because the math was too hard for a percentage in the extreme upper-90's, I've simply opted to exclude them.

Platina Ezel - New Trigger Calculations

Since I've been waiting for 3 hours to get into MTG Arena and it's still down, I decided to burn some time by writing this article.  Yes, I do math when I'm bored.

For those unaware, Platina Ezel's new effect is:

Blazing Lion, Platina Ezel 
[ACT](VC):[COST][Counter-Blast 1], and change this turn's first drive check to "look at two cards from the top of your deck, reveal one card from among them in your trigger zone, and call one card from among them to (RC)." If your soul has two or more grade 3 cards, change the second drive check of the turn and more as well.

Let's start with the basic odds.

Odds of hitting at least one trigger off of a normal Twin-Drive!!: 55.1%
Odds of hitting at least one trigger with no grade 3's in soul: 70.39%.
Odds of hitting at least one trigger with a grade 3 in soul: 80.88%

Odds of hitting two triggers off of a normal Twin-Drive!!: 10.2%
Odds of hitting two triggers with no grade 3's in soul: 17.22%
Odds of hitting two triggers with a grade 3 in soul: 30.36%

The big thing to notice here is that your odds of hitting two triggers is the biggest boon this card's selective trigger checking grants to you.  The odds are the same as hitting a single trigger or more on two independent Twin-Drives!!, which is why the number is so comically large.  If you are just aiming to hit a singular trigger, you should recall that the number for no grade 3's in soul is identical to the odds of hitting a trigger with OG CEO Amaterasu.  Obviously, one-to-passing Ezel is going to be a non-option.  Two-to-passing also isn't particularly safe (especially if fronts are involved), as the odds of the Platina player hitting two triggers are about the same as the odds of a normal Twin-Drive!! getting a single trigger.  Hopefully, being armed with this information will help to keep you aware of what the general risk level is with each decision.