# Magic Math – Magic Origins, Part 1

Magic Origins introduces a ton of interesting cards, and in this article I’ll cover several with recreational math. For the most part, this will take the form of straightforward hypergeometric probabilities, which have plenty of useful applications to Magic: the Gathering. As an illustrative example, suppose you have a 60-card deck with 24 lands and are interested in the likelihood of drawing exactly 2 lands in a 7-card opening hand. Since you’re drawing without replacement, that’s a hypergeometric probability! Likewise, cards like Gather the Pack or Talent of the Telepath read like textbook exercises on hypergeometric probabilities to me.

I won’t go into too much detail on the math behind it, but if you want to learn more, then you can check out this website. It contains explanations, formulas, examples, and a simple hypergeometric calculator. For the situation I sketched above, you can use that calculator by filling in 60 for “population size,” 24 for “number of successes in population,” 7 for “sample size,” and 2 for “number of successes in sample.” If you click Calculate, then the answer to the question, 0.269, will be listed after “hypergeometric probability.” In the remainder, I’ll refer to these kinds of probabilities in text as Prob[2 successes in 7 draws from a 60-card deck containing 24 successes]. All in all, the calculations are not all that complicated—you can run the numbers yourself, too!

# Gather the Pack

To understand how reliable Gather the Pack is, let’s start with a concrete question: Suppose we have a 60-card deck with 20 creatures. We remove one Gather the Pack from the deck, turning it into a 59-card deck, and put it on the stack. Of interest is the probability of hitting one or two creature cards.

Without spell mastery, we hit if we see at least one creature in the top 5. The easiest way to determine the probability of such a hit is by equating it to one minus the probability of a miss, i.e., of seeing exactly zero creatures in the top 5. In other words, it’s 1 – Prob[0 successes in 5 draws from a 59-card deck containing 20 successes]=88.5%.

With spell mastery, we hit the jackpot if we see at least two creatures in the top 5. We can obtain the probability in a similar fashion, but we have to additionally subtract the probability of seeing exactly one creature in the top five. So, it’s 1 – Prob[0 successes in 5 draws from a 59-card deck containing 20 successes] – Prob[1 success in 5 draws from a 59-card deck containing 20 successes] = 55.6%.

We can do the same type of calculation more generally for a 60-card deck containing a certain number of creatures. Here’s what comes out:

Now that we have this, we can wonder how many creatures a Gather the Pack deck needs. There is a natural tension between creatures and spell mastery, and the likelihood of having spell mastery is influenced by many factors such as which turn it is. This makes it tricky to give general recommendations, but nevertheless I’d suggest that you ensure that a turn-2 Gather the Pack, which you’ll be forced to play from time to time, should hit one creature at least 90% of time. This comes down to a minimum of 22 creatures. Beyond that, the optimal number of creatures really depends on the deck and format.

# Talent of the Telepath

Suppose you’re playing against a deck with 16 instants or sorceries, which is close to what many Abzan Control, Jeskai Aggro, and Mardu Dragons decks are playing in Standard. Also suppose that we have no knowledge of the cards drawn by our opponent. Then, the following holds:

• You’ll hit 0 spells in Prob[0 successes in 7 draws from a 60-card deck containing 16 successes]=9.9% of the cases
• You’ll hit 1 spells in Prob[1 success in 7 draws from a 60-card deck containing 16 successes]=29.2% of the cases
• You’ll hit 2 or more spells in the remainder, i.e., in 60.8% of the cases.

This makes for an expected value of 1.51 spells with spell mastery and an expected value of 0.90 spells without spell mastery. That’s okay, but not spectacular. If you would be able to reliably get, say, a free Hordeling Outburst and Lightning Strike for four mana, then it would be a good deal, but I feel the probability of missing is too large against a deck with 16 instants or sorceries, even if you’d always be able to guarantee spell mastery. Moreover, the average mana cost of instants or sorceries in these decks is relatively low.

Talent of the Telepath may still hold utility as a sideboard card for the Jeskai Tokens mirror match, where the number of hittable instants or sorceries, including 4 copies of the super-juicy Treasure Cruise, hovers around 22. In that case, the expected number of hits becomes 1.78 spells under spell mastery and there’s a 39.9% probability of hitting at least one Treasure Cruise, which seems good enough. But apart from a few fringe uses like these, I don’t expect Talent of the Telepath will be strong enough for the current Standard.

# Llanowar Empath

Suppose we have a typical 40-card Limited deck with 15 creatures. We remove one Llanowar Empath from the deck and put it on the stack. Then, assuming you scry any non-creature card to the bottom, you’ll hit at least one creature with probability 1 – Prob[0 successes in 3 draws from a 39-card deck containing 14 successes]=0.75.

Counting scry 2 on a 4-drop as approximately drawing 0.40 cards, Llanowar Empath’s ability is comparable to drawing 1.15 cards on average. Hence, it will usually be slightly better than Striped Bears or Shaman of Spring, though not by much. This leads me to conclude that it will be a fine playable for Limited, but that it’s nothing too special.

# Nissa’s Revelation

The value of this card depends on the exact composition of your creature base, but we can get some feel for the card by considering a specific situation for illustration: you play it in a Limited game when you have a 25-card deck with three 5/5s and seven 2/2s. If you scry any non-5/5 to the bottom, then the probability of revealing a 5/5 is equal to 1 – Prob[0 successes in 6 draws from a 25-card deck containing 3 successes]=57.9% and the probability of revealing a 2/2 is equal to Prob[0 successes in 5 draws from a 25-card deck containing 3 successes] * 7/20=17.3%.

This makes for an expected power/toughness reveal of 3.2. If we count scry 5 as drawing 1.8 cards, which seems like a reasonable estimation in the late-game when scrying a land to the bottom is almost the same as drawing a card, then Nissa’s Revelation is in expectation similar to a draw 5 that gains you a bit of life. At least in the specific situation that I sketched.

Based on this quick-and-dirty, yet reasonable estimation, I’d generally peg Nissa’s Revelation at a similar power level as Dragonlord’s Prerogative. It’s better in the Cobblebrute deck and worse in the Guardians of Meletis deck, but in general it should be a decent card if you’re interested in a 7-drop. It might even have Constructed applications—I’m hoping for some crazy Bearer of the Heavens/Rescue from the Underworld/Nissa’s Revelation combo brew.

Don’t miss the Part 2 where I show you how many Elves you should run with Sylvan Messenger, and how many Timberpack Wolves you can expect to see in each draft!

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