# Taking Chances: Probability and the Fall of Argenport

Hi, jez2718 here again with another article on using probability to inform deck building. In my last article I introduced the framework for calculating deck building probabilities, and looked at some basic questions you can inform via probability. In this article we’ll be building on that (so if you haven’t read the last article, I recommend reading it first) to look at evaluating a couple of strategies made relevant by spoiled Fall of Argenport cards.

# “Discard is consistency” and pre-boarding

To the consternation of many competitive players (including me) Scarlatch commented the other week that DWD intends to use the market as a stand-in for side boarding in official organised play events. Resheph and I are in the process of writing an article quantifying the relative impacts of sideboarding vs. markets in a best of three, but there are certain things that you simply cannot do with markets. Notably, the earliest you can play a market card is turn 4.

We’ve been seeing a bit of a dry spell for fast aggro the last few months, but anyone who remembers back to when Rally decks (or earlier, Jito decks) were prevalent on ladder will know that some decks can simply kill you by turn 4. So the market is not a reliable saviour against these sorts of decks. A possible solution to this is to pre-board against such matchups, i.e. running in your main deck powerful but narrow cards like Lightning Storm for a prevalent matchup.

The weakness of pre-boarding of course is it dilutes your deck and reduces consistency. Drawing a Lightning Storm vs Big Combrei or Unitless makes you feel silly. This is where discard effects can come in. I here use “discard” in a loose sense of anything that trades a card in your hand for another card or a powerful effect: so alongside looters such as Nocturnal Observer and similar cards like the recently spoiled Lumen Attendant, I also here refer to effects like Strategize and, importantly, Merchants. The motto “discard is consistency” highlights that these sorts of effects can boost the consistency of your deck by converting dead cards into gas. So we ask the question:

## How many discard effects do I need to get away with running situational cards?

Let us consider the specific case of Lightning Storm. Suppose that fast aggro decks are, say, 20% of the meta, and that if you don’t draw a Lightning Storm by turn 3 your deck struggles vs these decks. Suppose the other 80% of the time the card is basically dead.

To reliably draw the Lightning Storm by turn 3, you really need to be running 3 or 4 copies of the card (with probabilities of 32% and 40% on the play respectively). Let’s say you run 3, after all this is a sideboard card, so you don’t want to dilute your deck too much, and you might still want one in the market.

The question is, how many discard effects should I run so that I can be confident of not drawing fewer discards than Lightning Storms on turns say 2 through 8? The probability of this on turn N with D discard effects is given, in the notation of my previous article, by the formula:

$\sum_{i=1}^{3} \text{HG}(75,3,6+N,i,=)\text{HG}(72,D,6+N-i,i,<)$

Recall that the first term of this is the (hypergeometric) probability of drawing exactly i Lightning Storm, and the second term is the probability of drawing fewer than i discard effects given that we draw i Lightning Storm. Calculating this we find

 Probability of drawing more Lightning Storms than discard effects (on the play) with 3 Lightning Storm in deck Number of discard effects Turn 2 3 4 5 6 7 8 1 26.5% 29.1% 31.5% 33.7% 35.9% 37.9% 39.9% 2 24.2% 26.2% 28.0% 29.7% 31.2% 32.6% 33.9% 3 22.0% 23.5% 24.9% 26.1% 27.1% 28.1% 28.9% 4 20.0% 21.1% 22.1% 22.9% 23.5% 24.1% 24.5% 5 18.2% 19.0% 19.6% 20.0% 20.4% 20.6% 20.7% 6 16.5% 17.0% 17.3% 17.5% 17.6% 17.6% 17.5% 7 15.0% 15.2% 15.3% 15.3% 15.2% 15.0% 14.8% 8 13.5% 13.6% 13.5% 13.4% 13.1% 12.8% 12.5% 9 12.3% 12.2% 11.9% 11.6% 11.3% 10.9% 10.5% 10 11.1% 10.8% 10.5% 10.1% 9.7% 9.2% 8.8% 11 10.0% 9.7% 9.2% 8.8% 8.3% 7.8% 7.3% 12 9.0% 8.6% 8.1% 7.6% 7.1% 6.6% 6.1%

Looking at these numbers, it seems just running the four Jennev Merchants still leads to a pretty good chance of having a dead Lightning Storm. To really use this strategy effectively, you probably want to be looking at more like 7, 8 or more discard effects. Still with say 4 Strategize, and with the fact that we’ve ignored the possibility of redrawing hands with Lightning Storm or shipping them with Crests, this strategy could well be a viable way to fight against fast aggro without a sideboard. Note the trade-off however. With 8 discard effects, 3 Lightning Storm and a 20% fast aggro meta: 6.5% of the time you draw a Lightning Storm on time and hose fast aggro, 10.5% of the time you draw a mostly dead card and the other 83% of the time nothing much happens. So if you’re employing this strategy, the card has to really have an impact when you want it, and the decks you want it for need to be sufficiently common.

# Standards in Aggro

Last weekend, DWD spoiled the remaining four Standard/Tactic cards, and especially the Fire, Justice and Shadow ones raise an interesting question.

## How many of these should you run in Aggro?

Obviously, the combat trick side of these cards is great. I think the Fire one is especially important, as by the time it transmutes Aggro will typically be trying to push for those final points of damage. And it helps that these don’t reveal their presence with pauses until they transmute. However, since an Aggro deck wants to always curve out, running more depleted power is a risk.

Now, questions like this are deceptively complicated. There are a lot of factors that go in to determining what your power should look like. So though we can use the techniques in my last article to find a few relevant probabilities, we must remember these tell only part of the story. Another approach, which we’ll touch on briefly, is to run simulations to save us very tricky calculations.

## Some important probabilities

For simplicity, let us suppose our Aggro deck runs 25 Power: 4 Seats, 4 Banners, 0-8 Standards and the rest Sigils. Often a Stonescar deck might supplement this with a couple of Vara’s Favor, but for the purposes of curving out a Favor is very unlike even a depleted Power. Recent Rakano decks have been running exactly 25 sources, as did Sunyveil’s Stonescar at Worlds, so this isn’t unreasonable.

### Probability of having undepleted power on curve on turns 1, 2, 3

This probability is the probability of drawing at least X power on turn X, at least one of which is undepleted Power. We can count Seats as depleted (as if you drew an undepleted seat, you already drew a undepleted Power). On turn 1 Banners are depleted (Infernus is a bad card!), and are only undepleted on turn 2 if you drew another undepleted source. We’ll assume they’re undepleted on turn 3 however. So if we run S standards, we have 17-S undepleted sources in deck on turns 1 and 2 and 21-S on turn 3. Thus on turn X = 1,2 we calculate:

$\sum_{i=X}^{6+X} \text{HG}(75,25,6+X,i,=)\text{HG}(25,17-S,i,1,\geq)$

and on turn 3

$\sum_{i=3}^{9} \text{HG}(75,25,9,i,=)\text{HG}(25,21-S,i,1,\geq)$

where the first term is the chance of drawing exactly i power, and the second the chance that at least one of that power is undepleted. We find:

 Standards Turn 1 2 3 0 84.9% 78.6% 63.4% 1 82.8% 77.4% 63.3% 2 80.5% 76.0% 63.2% 3 78.0% 74.3% 63.0% 4 75.2% 72.3% 62.6% 5 72.1% 69.9% 62.2% 6 68.7% 67.3% 61.5% 7 64.9% 64.2% 60.7% 8 60.8% 60.6% 59.7%

### Probability of drawing 5 non-Standard Power and a Standard

This tells how often the standard will actually be worth it, since you don’t really want to be relying on your other standards to get your Tactics on line. The probability is relatively simple. On turn N:

$\sum_{i=5}^{5+N} \text{HG}(75,25-S,6+N,i,=)\text{HG}(50+S,S,6+N-i,1,\geq)$

 Probability of drawing 5 non-Standard power and a Standard Standards Turn 5 6 7 8 1 2.6% 4.0% 5.8% 7.8% 2 4.3% 6.6% 9.5% 12.9% 3 5.2% 8.1% 11.6% 15.8% 4 5.5% 8.6% 12.5% 17.1% 5 5.4% 8.5% 12.4% 17.1% 6 5.0% 8.0% 11.7% 16.2% 7 4.5% 7.1% 10.6% 14.7% 8 3.8% 6.1% 9.1% 12.8%

### Probability of undepleted Power on curve on turns 2 AND 3

This is made harder by the way Seats work. It can be calculated exactly, but the calculations are a bit too messy for this article. Instead we can estimate more easily with a simple simulation: we shuffle a deck of 75 cards (50 non-power, 4 Banners, 4 Seats, S Standards and 17-S Sigils) thousands of times, and count how often we have undepleted Power on curve. We can even incorporate the redraw rule into our shuffle if we please.

 Standards Probability of undepleted Power on curve turns 2 and 3 (estimate from 10,000 shuffles each) 0 53.4% 1 52.6% 2 51.8% 3 50.0% 4 49.1% 5 47.5% 6 45.0% 7 42.5% 8 40.2%

## Simulations in a goldfish bowl

A good metric for the quality of an Aggro list is the “average goldfish kill”, i.e. the average number of turns it takes for the deck to kill an opponent who does literally nothing. Finding the exact effect of running some number of Standards on this quantity would be a hell to calculate. But if you wanted, you could estimate this more easily by running simplified simulations. For example, one could consider a deck with an Aggro-like curve of vanilla X/X for X, 25 Power, some (0 to 8) of which are Standards that turn into “deal 4 damage” spells for 2, and then test a couple of thousand shuffles for their average goldfish kill for each amount of standards.

# Wrapping up

Set 4 is shaping up to be an interesting set, raising various questions for deck builders. Hopefully this article has shed some light on a couple of these questions. The full impact of Merchants in particular will be exciting, and keep your eyes out for an article by Resheph and myself on how markets compare to sideboards.

Finally, don’t forget that TGP hosts a “Casual Friday” tournament every Friday at 5 EDT!

Until next time,