The ultimatums are some very powerful cards in IKO limited, but only if you can actually cast them. This takes a look into the requirements on your manabase to be able to cast them somewhat reliably.
Methodology
If you don’t care about the math behind the numbers I’m about to calculate, feel free to skip down to the last section titled “The Takeaway”.
The hypergeometric formula is a very useful tool for calculating the odds of drawing a certain set of cards from your magic deck. It is expressed as:
Where:
N = Population Size (in limited this will be your deck size, typically 40)
k = Number of Successes in the Population (number of cards of the type you’re interested in drawing)
n = Sample Size (the number of cards drawn)
x = Target Number of Successes in Sample (the number of successes you are looking for in the sample).
Note: this gives the odds of drawing exactly x of a card. If we want to know the odds of drawing at least x, we sum P(N, k, n, x) from x = the target number to k. So the odds of drawing at least 7 lands from n cards using a 40 card deck with 17 lands would be:
In a 17-land deck, this equation will produces the following curve for odds of having drawn enough lands to a card that cost 7 generic mana:
To expand those out to 1–7 generic mana, we get the following curves:
Now, what about the odds of drawing lands that can produce the XXYYYZZ cost of an Ultimatum? If we have only basic lands in our deck, we can use the multivariate hypergeometric formula, given by:
Where:
N = Population Size (in limited this will be your deck size, typically 40)
n = Sample Size (the number of cards drawn)
k₀, k₁, … = Number of Successes in the Population for each type (number of cards of the type you’re interested in drawing)
x₀, x₁, … = Target Number of Successes in Sample for each type (the number of successes you are looking for in the sample).
So if we were to consider the following mana base:
5x Plains
7x Swamp
5x Forest
What are the odds we have drawn WWBBBGG when we have drawn exactly 7 lands?
For this scenario we use k₀ = 5, k₁ = 7, k₂ =5, x₀ = 2, x₁ = 3, x₂ =2, N = 17 and n=7
We can now combine these equations to find the odds of drawing at least enough lands to produce our desired mana. First we can take the odds of drawing a specific number of lands, then we multiply it by the odds of drawing a valid configuration of lands given that we’ve drawn that specific number of lands.
This ends up becoming the following:
This equation can be used for manabases including non-basics now.
Defining “Consistently”:
For the analysis below, I’m going define consistently able to cast as 85% of the time. This is approximately equal to the percentage of the time you will hit your 3rd land drop on turn 3 on the play. (Something I expect to be able to do, but something I’m not completely shocked if it doesn’t happen.)
So, what turn can we expect to consistently be able to play a card that costs only generic mana? From above, we know the following curves:
When we put a numeric turn on when we can “expect” to play a given we end up with the table below for generically costed 2–10 drops:
So, what turn can we expect to play an ultimatum consistently? That depends on how we build our manabase…
Some Example ‘Balanced’ Manabases:
Consider the following manabases, where the land balance is optimized towards casting the Ultimatum:
These are the odds of being able to produce Ultimatum Mana per cards drawn (using the equation derived in Methodology):
Importantly, this methodology does not consider the possibility of your lands entering the battlefield tapped. If the last land you’ve drawn to be able to hit ultimatum mana is a tapland, you will need to wait an extra turn cycle to play it. This should happen roughly proportionally equal to the number of taplands you have. So, the turns we can ‘expect’ to play the ultimatum with these manabases are as follows:
Some Example ‘Unbalanced’ Manabases:
Very few limited decks can support roughly evenly distributed 3 color manabases like the above. Often, you’ll be strongly two colors with a weak third. Consider the following manabases:
These are the odds of being able to produce Ultimatum Mana per cards drawn:
This then produces the following expected turns:
The Takeaway
I would recommend against playing an ultimatum in a deck with fewer than 19 sources and fewer than 6 pieces of fixing within those sources. (Depending on how much of that fixing is 2-color vs 3-color fixing)
Cycling and card draw can mitigate some of this, but you’ll want a lot. Tables shown below for various example manabases and what turn they are 85% likely to have found ultimatum mana. Note: This is the turn they are likely to have found ultimatum mana, as multicolor sources are typically tapped, there is likely to be a turn delay. This also assumes Evolving Wilds and Farfinders are used as late as possible (once you know what color source you are otherwise missing).
17 Sources:
18 Sources:
19 Sources:
20 Sources:
