Probability and Eggs

Hello all,

While playing Eggs, you can get yourself into a few sticky situations. In this article I will attempt to explain just how sticky those situations can be. Should you go for it, or will you likely end up with egg on your face?

Probability and Hypergeometry

First, let me direct you to this site: Hypergeometric Calculator. Getting accustomed to the use of this site while playing MTGO or getting a few ranges of probability for paper Magic will improve your game immensely.

Here are a few facts about the Modern format and probability:
1) While playing combo, you want to assemble two or more cards to win.
2) By deciding to keep an opening hand, you have one or more of those cards.
3) Most Modern cantrips function as an extra draw step for the immediate turn: [c]Gitaxian Probe[/c], [c]Serum Visions[/c], etc. [c]Sleight of Hand[/c] is an exception.
4) Some Modern decks, like Eggs and Storm, instead have engines that let you draw many cards.

So how do we use the Hypergeometric Calculator to play Magic? Well, let’s say we are playing [c]Ad Nauseam[/c] combo, and we have [c]Angel’s Grace[/c] and six mana but not the namesake piece. If our opponent has a three-turn clock on us, then we have three draw steps.
In the “Population size” space of the Calculator, we enter the size of our library. I’ll say 50.
Any one of our four [c]Ad Nauseam[/c]s is a success, so we enter 4 in the “Number of Successes in population” space.
We have three draw steps remaining, so we enter 3 in “Sample Size.”
Only one [c]Ad Nauseam[/c] will do the trick, so we enter 1 in “Number of Successes” and press “Calculate.”

The website spits out five new numbers, and the last of these is the likelihood that in three cards, we will draw one or more [c]Ad Nauseam[/c]. With no other factors, we have a 22% chance to win this game. Looking at it in long-range match terms (which you should when picking a deck), we will win one out of every five games that this situation comes up.

But that isn’t the fun part. Here, we are saying that we find the card or die. What if our clock is unknown? What if the [c]Splinter Twin[/c] player has three untapped lands at the beginning of our turn? Or what if Affinity has 7 artifacts, [c]Inkmoth Nexus[/c], and [c]Cranial Plating[/c], and one more artifact will kill us? Here we can decide what the cost is to try and go for it, and what percentage of the time we should win in these scenarios. Additionally, these scenarios come up very frequently.

Scenario #1: We need Ironworks
Often on turns three and four we will have three eggs and four mana available: [c]Chromatic Star[/c], [c]Chromatic Sphere[/c], and [c]Terrarion[/c]. Since each of these eggs replace their activation costs with added mana, we can sacrifice them all and have three chances to draw [c]Krark-Clan Ironworks[/c]. Again, we use our population size of 50 here, and all of our numbers in the previous example are the same: 4 successes, 3 chances, 1 success matters. Only one in five games will we find [c]Krark-Clan Ironworks[/c] with this method. A [c]Faith’s Reward[/c] can help us recoup our losses, but we can’t keep the cycle going without more mana. Here, without knowing that we are going to die next turn, it is only good to wait or have the value [c]Faith’s Reward[/c] turn. Continuing to cantrip is only good if we have [c]Open the Vaults[/c] and are on that plan.

Scenario #2: We need Faith’s Reward or Open the Vaults
In this scenario, it is turn four, and we have tapped out to play and resolve [c]Krark-Clan Ironworks[/c]. We have played a [c]Terrarion[/c], [c]Chromatic Star[/c], [c]Chromatic Sphere[/c], and [c]Ichor Wellspring[/c]. No amount of mana will be a problem, but we do not have either of our 7 combo pieces.
Population – 50
Number of Successes – 7
Sample Size – 4
Number of Successes in Sample – 1
We find here that we have a 47% percent chance to find [c]Faith’s Reward[/c] or [c]Open the Vaults[/c] from these four cards alone, and we will win half these games based on this probability.

What’s beautiful about playing Eggs, though, is that we can adjust our number of “successes” also to the cards that simply let us keep going. A land is a dud, but each egg piece allows us to lower our population and a new chance to “hit calculate,” as it were. Even [c]Reshape[/c] can find [c]Ichor Wellspring[/c] and likely be easily cast. In this scenario, while needing [c]Faith’s Reward[/c] or [c]Open the Vaults[/c], we give “success” the definition of the ability to keep playing, likely finding a win, instead of one of the seven cards outright. Usually this works. Let me acknowledge my failures in Math here and tell you that this is the section of the article that gets a little squirrelly.

Population – 50
Number of Successes – 21 (our eggs in deck – the ones I have mentioned on the table)
Sample size – 4
Number of successes in sample – 1

Here, our deck will only fail us and provide land, [c]Lotus Bloom[/c], [c]Mox Opal[/c], land one in ten games. We have a 90% chance to “win” with [c]Krark-Clan Ironworks[/c] and a handful of baubles provided that our opponent has no interaction. And let’s admit it; interaction will be tough against Eggs.

Application in Paper

Knowing that the probability of drawing 1 card in 50 with 3 chances comes up so much that I have used it twice in this article: roughly 20% chance of working. How do the numbers adjust, roughly, per Eggs in play and cards in deck?

Well, if it is cards in our deck, the probability is affected very minutely. With 51 cards in the deck, our 22.55% probability (from 50 cards in the deck) becomes 22.13%. With 49 cards in the deck, the probability is 22.98%. So you get half a percentage point per card in the deck.

If it is Egg cantrips, the probability is a little greater affected. If we have four draws on the board and 50 cards in the library (instead of the previously used 3 draws), or 22% moves to 29%. With two draws instead of three, our likelihood drops to 15%. Every egg cantrip is worth 7% at finding what we need! This shows how important it is to continue to develop your board and not sacrifice eggs randomly until you are ready to go off.

Probability is a fascinating thing to explore while playing Magic: The Gathering, and it obviously has many applications outside of playing Eggs. If you want to improve your game, consider where these calculations can be used with your deck. Should you [c]Thoughtseize[/c] this combo piece or that one based on the clock you have? If you have looked at your opponent’s hand earlier in the game, what are the chances he has drawn a piece of interaction? Should you go for it? It’s a beautiful study, and you will win more by exploring it.