A turn-two double pip wants fourteen of your seventeen lands

The answer key already exists

Ask a draft table how many red sources a double-red two-drop wants and you will get a shrug and a number that sounds right. The number is already published. Frank Karsten's 2022 source tables, computed for a 40-card deck with 17 lands, list the minimum colored sources for every pip requirement a Limited card can pose: a single pip at one mana takes 9 sources, and a double pip you want to cast on turn two takes 14 of those 17 lands.¹

Fourteen is not a typo and it is not conservative padding. It is what the arithmetic returns when you ask for the color often enough to cast the card on curve, and it is why "just splash it" and "run a couple more Islands" are answers to a question that has an exact one. Karsten defines a manabase as able to consistently cast a card when there is at least an 89 + M percent chance to draw the colored sources it needs by the turn it costs, on the play, conditional on having drawn enough lands to pay for it. That target climbs with mana value, from 90 percent for a one-drop to 96 percent for a seven-drop.¹

The formula you can rerun

The engine is one line of combinatorics. Drawing sources out of a deck is sampling without replacement, so the count of on-color sources in your opening cards follows the hypergeometric distribution: for a population of N cards containing K successes, the probability that n draws contain exactly k successes is P(X = k) = C(K,k)·C(N−K, n−k) / C(N, n).² Map the four letters onto a manabase and nothing is left abstract: N is the deck (40), K is the number of lands that make your color, n is the cards you have seen by the turn in question, and k is how many on-color sources you actually want in hand. The quantity a manabase turns on is never a single k but a tail: the chance of drawing at least the pips you need, P(X ≥ k), which is the sum of the formula from k upward.² Players who keep a hypergeometric calculator open at the deckbuilding table are computing exactly this: population is the deck, successes are the sources, draws are the cards seen.³

Work one cell by hand so the table stops being a black box. Take a single-pip one-drop with 9 sources, and ask for the color in your opening seven on the play. Here N = 40, K = 9, n = 7, and you want k ≥ 1, so it is easiest to subtract the miss: the chance of drawing zero of the 9 is C(31,7)/C(40,7) = 0.141, which makes P(X ≥ 1) = 1 − 0.141 = 0.859, or 85.9 percent. Do the same for a double pip you must hit twice, a turn-two WW two-drop run on 14 sources: P(X ≥ 2) = 1 − [C(26,8) + 14·C(26,7)] / C(40,8) = 86.0 percent. One card, one number, reproducible in any calculator to the tenth of a point.

Why it is a curve, not a threshold

The table hands you a minimum, but the honest object underneath is a curve. Below is the bare hypergeometric probability of drawing the pips you need against the number of on-color sources, for the two requirements that decide most decks: a single pip cast on turn two, and a double pip cast on turn three. No mulligan, no card selection, so a reader can reproduce every point.

The single-pip curve is nearly flat by the time it matters: past 9 sources each extra land buys two points, then one, then a fraction. The double-pip curve is a different animal. It crosses 90 percent only at 14 sources on turn three, and every source below that is expensive, moving the number by seven or eight points at a time. "About 9 sources" is right for a single pip and disastrous for a double: at 9 sources a turn-three WW spell is a 66.7 percent proposition, closer to a coin flip than to a plan.

The 90 percent line the table chases is a choice, not a law, and it is worth saying so out loud. Karsten proposed it more than a decade ago, in the original version of this analysis, on the reasoning that access to your mana nine games in ten is a defensible reading of "consistent."⁴ It is a judgment call, and a reader who wants 85 percent or 95 percent can read their own line straight off the curve. What is not a judgment call is where the curve sits, and that is the part the formula settles.

What a splash actually costs

A splash is a small number of off-color sources bought to cast one or two cards, and its price is exactly the tail probability at that small number. Karsten's own reduction makes the accounting simple: splashing a card into a deck whose other lands all make your main color is equivalent to running that card on the splash's source count alone, so a single-pip splash is priced by nothing but how many sources you gave it.¹ Run the tail for a bare single-pip splash, on the play, and the cost is stark.

Give every number the yardstick a drafter already owns. A full on-color single pip runs around 9 sources and clears roughly 90 percent on curve, the consistency of a card you would never think to cut. A typical two-to-three-source splash lands between 40 and 67 percent to be online by the turn you want it. You are more likely than not to have the splash only from three sources, and already by turn three, where it is there 54.5 percent of the time; one source never reaches a coin flip, and two sources edge past half only around turn six. The card is not a free inclusion on your curve; it is a card you will be holding in a dead hand a third to a half of the time you want it.

The same math punctures the three-color manabase that splits 17 lands into a tidy six, six, five. An independent Limited analysis running the identical model puts a six-source color at only 71 percent to appear in the opening hand and 81 percent by turn three, and confirms the nine-source figure at 86 percent in the opening hand.⁵ Those match the bare computation to the tenth of a point: 6 of 7 is 71.1 percent, 6 of 9 is 80.8 percent, 9 of 7 is 85.9 percent. Two colors at six sources each is not a balanced base; it is two colors that each fail you a fifth of the time in the first hand. And gold cards compound the tax: because you need both colors present, Karsten raises each requirement by one, so a two-color card sitting at 90 percent per color is only 90 × 90 = 81 percent to be castable on curve.¹

Where the calculator and the table disagree

Plug 9 sources, 7 cards, and a 40-card deck into a calculator and you get 85.9 percent. Karsten's table lists 9 sources as sufficient for a single-pip one-drop at a 90 percent target. The four-point gap is not an error in either place; the two numbers answer slightly different questions, and keeping them apart is the difference between using the model and misquoting it. The calculator gives the floor: the raw chance of drawing the color, counting every mana-screwed game as a failure. Karsten's published minimum adds two refinements his simulation makes explicit in code. The deck is modeled as good lands, other lands, and spells; a game is counted only when total lands drawn already cover the turn; and hands are kept or shipped under the London mulligan, keeping seven-card hands with two to five lands and digging toward three.⁶ Both refinements raise the probability of having the color given that you can pay the generic cost, which is the question a player cares about.

Rerun his conditional definition and the gap closes to about one source. Conditioning on having enough lands, that same one-drop cell reads 87.0 percent before any mulligan, and the London mulligan carries the last three points to his 90 percent target. Turn one is the single case that leans on the mulligan; by turn two the requirement clears on its own, with a single-pip two-drop at 9 sources reaching 92.5 percent against a 91 percent target, and a turn-two double pip at 14 sources landing at 91.5 percent. Karsten's counts reproduce from the bare distribution plus his two stated refinements to within roughly a single source: the recommendation is open, checkable, and honest, not a number handed down.⁶

The one thing this model does not give you is a current-metagame win rate. It would be satisfying to close with a line like "decks at N sources win X percent more," and the public data to attempt it exists: 17lands publishes aggregated, anonymized, per-set game and draft files from opted-in Arena players, with a known skew toward more engaged players.⁹ But a matured, verifiable win-rate cut that isolates color-consistency from everything else a manabase does is not on the table today, and a single small sample would be too fragile to trust. The model needs no such crutch. It is the field standard other builders already reach for when they want the odds of the mana working out,¹⁰ it is reproducible on a calculator, and it stands on its own. The number you had a feel for is computable. Compute it, and price your splash before you register it.