What are the Odds of a Black Lotus in Magic 30th Anniversary Boosters?

Karsten crunches the numbers!

On October 4, Wizards of the Coast announced a new product: 30th Anniversary Edition. It’s described as a “commemorative, collectible, non-tournament-legal product celebrating 30 years of Magic” that “lets fans and collectors experience some of the most iconic elements of Magic’s early years.”

Each display of the 30th Anniversary Edition will contain four 15-card packs with cards from Limited Edition Beta in their original rarities. The price tag is $999, or nearly $250 per pack. In this article, I will answer several questions that I found interesting from a mathematical perspective.

What is the Probability to Open a Black Lotus?

To figure out the probability of pulling Black Lotus from the 30th Anniversary Edition, we first need to determine the total effective number of Rares in the set. The original Limited Edition Beta contained 117 Rares, including Crusade but excluding basic Island. According to the 30th Anniversary Edition announcement, no rarities have been shifted, but six cards were removed to meet modern standards. Four of those are Rare: Contract from Below, Darkpact, Demonic Attorney, and Crusade—this leaves us with 113 Rares.

However, each of the ten dual lands (Volcanic Island, Tundra, etc.) will appear twice as frequently as any other non-dual land Rare. From a probabilistic perspective, this is equivalent to adding ten dual lands to the set, bringing the total number of Rares to 123. This means that each Rare should act like a random, independent draw from 123 cards, with dual lands appearing twice.

According to the announcement, each pack contains one Rare and one retro frame slot, which contains a Rare approximately three out of ten times. Taking all this into account, the probability P[No Lotus] that neither the Rare nor the retro frame slot features a Black Lotus is 122/123 * (0.7 + 0.3 * 122/123) = 0.9895. This means that the probability to open at least one Black Lotus in a single pack, 1 – P[No Lotus], is 1.05%, and the probability to open at least one Black Lotus in four packs, 1 – P[No Lotus]^4, is 4.15%.

As a long-run average, disregarding all other cards you’d open along the way, you’ll crack 1 / (1 – P[No Lotus]^4) = 24.1 copies of 30th Anniversary Edition before seeing your first Black Lotus, which comes down to more than $24,000. For that price, you may be able to purchase an actual Beta Black Lotus, albeit in poor condition. Since a Magic card weighs about 0.064 ounces (1.8 grams), a $24,000 card is worth more than its weight in gold, rhodium, platinum, heroin, meth, cocaine, LSD, and even plutonium!

What is the Probability to Open Two Pieces of the Power Nine?

Because packs contain both a Rare slot and a retro frame slot, you might hit the jackpot. As the announcement states, “some packs will contain two Rares, and some will even contain two pieces of the Power Nine!”

But how likely is that? Well, let’s run the numbers. The Power Nine is comprised of all five Moxen, Black Lotus, Ancestral Recall, Time Walk, and Timetwister. The probability that the Rare slot contains Power is 9/123 and the probability that the retro frame slot contains Power is 0.3 * 9/123. Multiplying both probabilities yields 0.16%. So approximately one out of every 623 packs will contain two pieces of the Power Nine

What is the Expected Number of Dual Lands?

Since dual lands appear twice as often as other Rares, the expected number of dual lands in one pack is 20/123 + 0.3 * 20/123 = 0.21. This means that in expectation, every four-pack 30th Anniversary Edition contains 0.85 dual lands. (Numbers are rounded to two decimal places.)

How Many Dual Lands and Pieces of the Power Nine Will You See?

I will define an “appealing Rare” as either a dual land or a piece of the Power Nine. This is an abstraction, as the set also contain iconic rares such as Shivan Dragon, Time Vault, Wheel of Fortune, Chaos Orb, or Mana Vault. The basic lands, tokens, Sol Ring, and several of the Commons and Uncommons will appeal to various players as well. But based on the market value of original Beta cards, I believe that the dual lands and Power Nine are the real standouts that you’re hoping to open the most.

Given this definition, the expected number of appealing Rares in one pack is 29/123 + 0.3 * 29/123 = 0.31. This means that in expectation, every four-pack 30th Anniversary Edition contains 1.23 appealing Rares. If card value is concentrated purely in these cards and there is no difference between frames or between appealing Rares—none of which is true—then you’re paying $814.83 per appealing Rare. I emphasize that this is a simplified overestimation, but it offers a first ballpark value.

However, that’s just the expectation. Sometimes you open sweet packs; other times you open unappealing ones. How about the distribution?

In a single pack, you’re 71.0% to open zero appealing Rares, 27.3% to open one appealing Rare, and 1.7% to open two appealing Rares. By convolution, the distribution for a four-pack 30th Anniversary Edition is described as follows:

  • 0 appealing Rares: 25.4% probability
  • 1 appealing Rares: 27.8% probability
  • 2 appealing Rares: 25.0% probability
  • 3 appealing Rares: 8.5% probability
  • 4+ appealing Rares: 13.3% probability

So roughly one in four 30th Anniversary Editions contain no appealing Rares. If you’re unlucky, you might pay $999 and open Living Artifact, Thoughtlace, Animate Wall, and Sunglasses of Urza.

How Many 30th Anniversary Editions do you Need to Open for a Complete Set?

In 1993, Collectors’ Edition was released. This non-tournament legal set included one of each card in Beta in gold borders, and it originally retailed for $50.

Inspired by Collectors’ Edition, suppose that you would like to collect one of each non-token card in 30th Anniversary Edition. So you want one copy of every single common, every single uncommon, and every single rare. You don’t care whether you have a card in normal or retro frame—both are equally good. Flying in the face of wisdom, we’ll keep opening fresh packs of 30th Anniversary Edition until we have a full set. 

So even if we have one copy of every card except for Thoughtlace, we’re just going to keep cracking packs until we open our first copy! We’ll never trade, buy, or sell singles. This may not be a financially prudent approach, and we might open a thirteenth duplicate copy of another rare before finally seeing that first Thoughtlace, but it makes for a fun thought experiment.

To determine how many packs we need to open to collect a full set in this fashion, I coded a quick simulation. The outcome is that in expectation, as when starting from scratch, you’ll have to open 496 packs to get a first copy of each card. This means purchasing 124 copies of the 30th Anniversary Edition for a grand total of nearly $124,000. Cool. The upcoming Magic World Championship pays out $100,000 to the winner.