## Articles

Hello, I’m Joseph Bogaard and I’ve been playing Magic for most of my life (16 years). I started playing Vintage around 2004 when I met a teacher at my high school that played Vintage (he had a full set of Power), and I was hooked immediately. I top-8ed several Vintage tournaments including Vintage Worlds 2011, and I’ve had a blast the entire time whether it were casting **Tendrils of Agony** or attacking with **Tarmogoyf** and **Dark Confidant**.

Right now I am an MD/PhD student in the third year of my PhD studies, which means I now have more time to play Magic! I was very excited, finally, to get to sling **Ancestral Recall** on MTGO, and even more excited that I get to write about it and record some sweet Vintage action.

For my first article, I thought I would write about the first major decision that occurs in every Magic game: the mulligan.Well,I chose to play or draw before that, but come on… the “right choice” is nearly always to play first, especially in Vintage. When **Tolarian Academy** and **Tinker **were printed and legal in Standard, the joke was the “early game” was the die roll and shuffling up. The “mid game” is mulliganing, and the “late game” was turn 1. Given that Tolarian Academy and Tinker** **are legal in Vintage (albeit restricted) and not even the most broken cards in the format, it’s not surprising then that this saying still has some truth for Vintage. While the “late game” may not really be turn 1, mulligans still play a big role in a player’s long-term win percentage. There is even a deck devoted to abusing the fact that if you mulligan properly, you can start the game with a single card** (Bazaar of Baghdad**) in your hand greater than 93%** of the time. I feel of all of the aspects of the game, mulliganing is something that isn’t talked about or thought about enough, which means that nearly all of us could improve our gameplay and our win percentage by improving our mulligan decisions.

Mulligan decisions are an excellent example of lenticular design/complexity. For those not familiar with the term “lenticular design/complexity”, it refers to creating cards whose complexity is variable depending on the skill and experience level of the player. A clean example is **Festering Goblin**. Beginning players will see a B 1/1 that has a bonus when it dies, and they are happy. If, during the game, their respective opponents attack with a 2/2 into their Festering Goblin, they won’t block and trade because they are playing it as a vanilla 1/1 with a bonus at the end when it dies. Eventually, they will block the 2/2 with Festering Goblin to protect their life total. Then when the ability goes on the stack, they will choose an X/1 creature an opponent controls to kill. They do this because at their current experience level they tend not to think about a causality, therefore don’t connect the 1/1 creature and its death trigger, and don’t see that their Festering Goblin can actually trade with the opponent’s 2/2. Somewhere down the road, as they play more and more games, they will make the connection.

Deciding whether or not to mulligan has similar, invisible complexity. When a beginning player learns about mulligans, it’s something they are hesitant to do. Unless they have 0 lands, they would rather keep their opening 7 (because 7 cards are more than 6 cards). Eventually, they learn the rule of thumb that it is correct to mulligan 0, 1, 6, or 7 land hands. As they continue to get better and have more experience, they will start to consider mulliganing hands that have a good mix of lands and spells for example,if they don’t have a card that costs less than 5 in a limited game, or if they have 4 forests and 3 red spells. Fundamentally, the choice of whether or not to mulligan is about deciding if you will have a better chance of winning with a smaller random hand than what you have currently. Because your new hand is unknown at the time you make the decision, it is a difficult decision to make, especially because it can leave a player with such a bad feeling when one mulligans a mediocre hand into a worse hand. The decision needs to be based on understanding your deck, the matchup, and statistics.

Finally, and I want to stress this point because it is a hard realization to come to, mulligan decisions are either correct or incorrect independent of how good or bad the new hand is without regard of the outcome of that individual game. Try and keep this in mind as you make your mulligan decisions because while the overall goal of a mulligan is to increase your win percentage, it’s important to be results oriented over a long period of time, not over a single round or a single tournament.

**Understanding your Deck –**

The first step in making the correct choice is to understand your deck.What is your game plan, how do you win, and how does your plan change when you are on the draw compared to the play?In most constructed formats, decks usually have a single defined strategy most generally described as aggro, control, mid-range, or combo. While these groups can be broken up more than that, these terms are broad enough that you can meaningfully classify nearly any deck into one of these four big broad categories.Occasionally, you do have a deck that’s a true combination of two of these, like the Pod Combo decks in Modern, they have a combo, but a lot of games they just play mid-range and grind out opponents.Blue Vintage decks often feel like they also have a dual nature because even the control decks can cast a Tinker on the first turn and put **Blightsteel Colossus** into play. But categorizing your deck helps you make the correct mulligan decision because it lets you understand your deck’s goal, which gives you a metric to measure your hand. If you are playing a burn deck (Aggro), your goal is to deal 20 as fast as possible. Depending on the format,burn decks play very few if any repeatable sources of damage so it is simply a matter of counting how much damage you have and how much you need to draw. For a burn deck, 5 lands (none of them man lands), 2 spells (7 points of burn) is really bad; you need to find a way to draw 13 more points of burn, which is at least 4 burn spells in most formats. Conversely for a control deck, 5 lands 2 spells (say a sweeper and a cheap removal spell) is a more keepable hand.

For an aggressive red deck with 12+ one drop and two drop creatures, when you get an opening hand with 4 lands and no one or two drops, that is a pretty terrible hand. On the other hand, if you’re playing midrange or control, you are unlikely to be playing any one drop creatures and not that many two drop creatures. When you fan open 4 lands &no one or two drops it’s average or maybe even a good hand, especially on the play. Depending on the average speed of the format, it may be a questionable keep on the draw.

For a combo deck, you are asking yourself the following questions: can I assemble my combo right now? If not, how much help do I need and what are the odds that I draw the cards I need?Taken to extremes a deck like dredge can be considered a combo deck where the combo is Bazaar of Baghdad and you rarely care what your opponent is doing. The mulligan decisions for dredge game one are simple – mulligan any hand that doesn’t have a Bazaar of Baghdad. Even games two and three, it’s still all about having a Bazaar of Baghdad, but you are also hoping for other lands and some of your sideboard cards to help offset opponent sideboard cards as necessary.

Here is a good example. Imagine you are playing legacy and you start with this opening hand:

First of all, what deck are we playing? Based on this hand it could either be UW Miracles or Stoneblade, as all of these cards are almost always found in both decks. For this example, let’s say it is a UW Miracles and that we are on the play. We don’t have a **Brainstorm** or a **Sensei’s Divining Top** for manipulation, but we have **Jace, the Mind Sculptor** if we make it to turn 4, and a **Swords to Plowshares** and **Karakas** to help us get there. We have 2 blue cards to pitch to **Force of Will**, and a **Vendilion Clique** to disrupt and potentially pressure our opponent. This to me is a strong hand in the dark.

**Understanding the Matchup –**

The next step is to understand the matchup.How should the evaluation of your deck and hand change based on your opponents deck and game plan? Some decks, like dredge, don’t need to consider what the opponent is doing, but for other decks it can be the difference between a great hand and an awful one. Without getting into too many specific examples (those will come last), the quintessential example of this was Pro Tour Yokohoma 2007 (Time Spiral Block) where the correct play was to keep an all land or 6 land **Prismatic Lens** hand in the control mirror because the most important factor of that matchup, like in many control-on-control matchups,was to hit all of your land drops! However, there was an aggressive mono red deck in the format and keeping a 7 lander against mono red was terrible.

Here is a Modern example; you are playing Affinity. It is game 3, and you are on the play with a pretty unremarkable opening hand of:

You don’t have a finisher like **Cranial Plating** or an **Arcbound Ravager**, no card draw, no **Etched Champion** to ignore his removal, and a lot of mana for a deck that doesn’t need much to operate optimally. So it’s a mulligan, right? Against an unknown opponent, this is absolutely a mulligan. This hand isn’t fast enough to race any decks in Modern, which is what this aggressive deck is typically trying to do. But it’s game 2 so it’s not an unknown opponent. So would you keep it against a pretty standard RWU control deck [Add link to Random RWU list] that is playing **Snapcaster Mage**, **Restoration Angel**, **Path to Exile**, **Lightning Bolt**, **Lightning Helix**, **Electrolyze**, **Mana Leak**, **Cryptic Command**? That’s a pretty hateful deck, and it’s very difficult for affinity to go all in on cards like **Cranial Plating** or **Arcbound Ravager** with any expectation of success. But with **Spellskite** to protect your other creatures and enough mana to cast any spell you draw, that hand gives you the opportunity to gain some time and card advantage in a matchup that ends up being quite a grind. To put it another way, what 6 card hand can you expect to get that’s going to have a better win percentage against RWU than your starting 7? Maybe something with the potential for a fast poison kill, but that’s going to be hard to pull off against 14-16 spot removal spells and counter magic meaning you have to a) mulligan into a great 6 and b) have your opponent make a mistake and keep a bad hand. No thanks.The starting 7 seems fine to me.

**Statistics – **

Having a solid grasp of probability and statistics or at least the ability to make quick odds calculations can be very helpful for players, and not just in their mulligan decisions. Probability and statistics is the only math anyone really needs, and it’s something you use every day even without thinking about it. Using statistics is how we calculate the correct probability of an event happening. For the rest of the article I will be going through how to use statistics to determine the probability of an event occurring. For those who are unfamiliar with statistics, the important concepts for this article are:

- The difference between independent and dependent events
- Compound events (or multiple events)
- Set theory
- And permutation and combinations

Khan academy has some great videos at https://www.khanacademy.org/math/probability for anyone who is interested in learning more about these concepts.

Using these concepts we can calculate the probability that my opponent has a **Force of Will** in his opening hand.Or the chance he has both a **Force of Will** and a blue card to pitch to it, more importantly, the probability that he has **Force of Will**, a blue card, and at least one mana source to make it a keepable hand. It’s impossible to calculate the exact percentages without having your opponent’s decklist, but if you are willing to make some assumptions and live with “close enough”, you can get a reasonable estimation.

You are piloting a combo and want to go off turn 1 on the play. **What are the odds that my opponent has a Force of Will in his opening hand?** If we assume he is playing 4, the percentage change of having 1 or more **Force of Will** cards in hand is 1 – % chance of having exactly 0 in the opening hand. So the chance of NOT having a **Force of Will** is:

So the chance that my opponent has at least 1 **Force of Will** is 1 – .6 = .4 or 40%. This is the chance of starting with at least one of any 4 ofs.

For comparison, if you were only playing 1 **Force of Will** (or maybe something that is restricted like **Black Lotus**), the chance of NOT having that card in your opening hand would be:

So the chance of starting with a singleton is 1 – .8833 = .1167 or 11.67% here is a shortcut for singletons. If you look,almost all of the numerators and denominators cancel, leaving you with 53/60 = 0.88333333333. With 4 ofs, only 54, 55, and 56 cancel leaving you with a little harder math.

Ok, so 40% of the time my opponent has at least 1 **Force of Will**, but some percentage of the time he had exactly 1 **Force of Will**. With only 1 **Force of Will**, it’s also possible he was dealt a second blue card. Here are the odds of having exactly 1 **Force of Will**.

Now some of you are going,“Huh,why is it times 7”? Well, if you assume there are 4 in your deck &you draw your **Force of Will** first then for the rest of the draws you want to draw any card other than the remaining 3 **Force of Will**. If you then assume the **Force of Will** was drawn second, you end up with the same function as when you drew it first.Multiply the original function by 7 to get an accurate measurement of drawing exactly 1 **Force of Will**. So about 33.63% of the time, your opponent will get only 1 **Force of Will**. Now, let’s say you draw 1 **Force of Will**, but no other blue cards. This is where the math starts to get complicated. To do this accurately you need to know how many blue cards are being played. To make life easy, I’m just going to pick a few numbers based on Legacy and Vintage decks so you can get a feel for the range. I’ll use 16 as the low, 20 as the middle and 24 as the high.

So with 16 total blue cards, there is a 7.31% chance you would be dealt an opening hand with exactly 1 **Force of Will** and no other blue cards, compared to the 33.63% chance of being dealt exactly 1 **Force of Will**. With 16 blue cards, of the 33.63% of total hands that have exactly 1 **Force of Will**, 7.31% of total hands will have 1 **Force of Will** and no other blue card. To put it another way, of the 33.63% of hands that have 1 **Force of Will**, 7.31/33.63 = .2173 or 21.73% of the time you will have no blue card when there are only 16 blue cards in your deck. As you can see, going up to 20 blue cards cut this in half, and going up to 24 blue cards reduces the chance even further.

Conclusion: Only a small percentage (between 5% and 10%) of the time will he have no blue card to go with his **Force of Will**, reducing the amount of time he has a **Force of Will **he can cast to 36%-38%. Remember this is only the starting 7, so it’s useful for turn 1 and changes quickly as soon as more cards are drawn.

Theoretically, about 40% of the time your opponent is going to have a **Force of Will** in his opening hand and between 36% and 38% of the time he will have a blue card to pitch to it. But there is one thing we are forgetting; the hand still has to be keepable. Unfortunately, what is and is not a keepable hand is so hard to define (I’ve spent more than 2500 words so far going over it) that determining a meaningful probability is almost impossible. We could arbitrarily define what a keepable hand is based on the number of lands it has, but that adds a third card type (**Force of Will**, land, other) that we are trying to measure, which adds a lot of extra computation.

For example, if we arbitrarily say is that the only unkeepable hand is one with 0 lands and if we say that our deck has 16 lands, 4 **Force of Will** and 40 other cards, what are the chances that we have at least 1 land and at least 1 **Force of Will**? To do this without simulation, we need to break up the question. Here are the probabilities we have to calculate by hand to get this right:1. The probability of 1 Force of Will

1. The probability of 1 Force of Will

2. The probability of 1 Force of Will and 0 land

3. The probability of 2 Force of Will

4. The probability of 2 Force of Will and 0 land

5. The probability of 3 Force of Will

2 is a subset contained in 1, so 1-2 is the probability of 1 **Force of Will** with at least 1 land. Repeat and add the remaining probabilities together and we have our answer. We already calculated 1, and the math on 2 is the same as having a **Force of Will** and no blue cards, and 3-8 are more of the same but more complicated and a lot of work.

Instead, let’s just calculate the chance of having a **Force of Will** based on reduced hand size. If your opponent has 1 land in hand, what are the odds that at least one of the other 6 cards is a **Force of Will**? It’s not the same, but it has the added benefit of being a useful calculation to keep in mind for mulligans.

Here is the calculation for 6 cards and none of them are **Force of Will**:

So the chances of a **Force of Will** in a 6 card hand is 35.146%, a 5 card hand is 30.006%, a 4 card hand is 24.679%, and a 3 card hand is 19.005%. So realistically the chances of your opponent having a **Force of Will **on the first turn is lower than 35.146% because you have to have at least 1 land to keep. Calculating the exact chance of having **Force of Will** requires the number of lands in the deck to correctly calculate the chances of having 1-4 lands in his hand. If we assume you are playing 25 lands/mana sources, the chances of having exactly 1 land are:

The chances of having exactly 2 lands are:

Why the 21 instead of 7? Because there are 21 different orders to order drawing 2 lands and 5 non-lands (or nCr where n is 7 and r is the number of lands)

The chances of having exactly 3 lands are:

And the chances of exactly 4 lands are:

Just to show that it all adds up to 1, exactly 5 lands are:

Add them all together and they equal exactly 1. The chance of having 1, 2, 3, or 4 lands is (0.10507+0.25217+0.31182+0.21438) = .88344

So if we assume you will keep 1-4 land hands and you are playing 25 lands, the chances of having a **Force of Will** is

26.07%, which is pretty different from the 40% calculated earlier. As you decrease your land count, that chance will rise. Using a spreadsheet to speed up the repeat calculations, I calculated that for 21 land/mana sources the chance is 27.66%, which is a pretty small change.

Whew, that was a ton of math and theory, and we haven’t even looked at a single vintage hand! I understand that statistics isn’t everyone’s forte or even something everyone is interested in learning about, but what I hope everyone can get out of this are 2 things:

- Calculating the real probabilities aren’t quite as simple as they initially seemed; and
- Once someone else has done the math for you, it is very easy to integrate it into your thought process. Instead of wondering “what is the chance my opponent can
**Force of Will**my turn 1 tinker?” You will know it’s somewhere between 26 and 28% depending on how many lands/mox he/she is playing.

So next time I will go over some hands and walk through my thought processes. But before I go, I’ll leave you all with a hand to analyze.

Now, this hand is hardly unique to a single Vintage deck, so consider:

- What deck you might be playing;
- Whether you are on the play or the draw; and
- How does your opponent’s deck choice, if known affect this keep?

Next article I will go over this hand and many more. Until then!

Joseph

MuseofAnger on MTGO

@jdbogaard

** If you do the math on **Bazaar of Bagdad** without **Serum Powder** mulligans, it is 86.50442%, but **Serum Powder** makes the math extremely difficult and time consuming. Instead a simple program that mulligans until it finds **Bazaar of Bagdad** using **Serum Powder** mulligans can fairly accurately estimate the actual probability with 10,000 – 20 million iterations. Last time I did this, it was around 93%.

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