Mana Base Probabilities

March 28, 2012

Mana Base Probabilities

Studying the probabilities of the game can teach us a lot about mana bases. When you build a mana base you are trying to maximize the probability of being able to pay the mana costs of your spells. For Magic, and most card games, we can apply what's called a Hypergeometric distribution. Wikipedia says, "the hypergeometric distribution is a discrete probability distribution that describes the probability of k successes in n draws from a finite population of size N without replacement." Huh? I'll explain in Magic terms. A Magic game follows this model because it is a case of sampling without replacement. Your pool, or population, is your deck, and you sample (draw) some number of cards. Then you can calculate the probability of drawing one or more cards of a certain type. For example we can answer the following question: If you have four Lightning Bolts in your deck then what's the probability of drawing one in your opening hand?

The Hypergeometric formula takes in four parameters. Here's what each looks like in our example:

- The population size. The deck size. 60 in this case.
- The number of success states in the population. A success is a Lightning Bolt so four.
- The number of draws. We draw seven in our opening hand.
- The number of successes. This means how many do you have to get in your hand for it to be considered a success. In this case it's one.

To crunch the numbers you can consult the formula, it's pretty easy as long as you know what a binomial coefficient is (Wikipedia can teach you that too), but there are also calculators available to save you the trouble.

So what's the answer? The chance to draw exactly one Lightning Bolt in your opening hand is about 33%. What might be more interesting to a player is the chance to draw one or more Lightning Bolts. We can calculate this with a cumulative probability which just means adding up the chance to draw exactly one, exactly two, exactly three, and exactly four. The chance to draw one or more Lightning Bolts in your opening hand is 40%. This of course applies to more than just Lightning Bolts.

There's a 40% chance to draw at least one copy of a four-of in your opening hand, but what's the chance to get one by turn two? Turn five? What if I'm interested in drawing one of eight cards? I put together a chart which has the probabilities to draw at least one of a card for which you have X copies in your deck by turn T.

Chance of drawing a card by a given turn

	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18
T1	12	22	32	40	47	54	60	65	70	74	78	81	84	86	88	90	92	93
T2	13	25	35	44	52	59	65	71	75	79	82	85	88	90	92	93	94	95
T3	15	28	39	49	57	64	70	75	79	83	86	89	91	93	94	95	96	97
T4	17	31	43	53	61	68	74	79	83	86	89	91	93	95	96	97	97	98
T5	18	34	46	57	65	72	78	82	86	89	91	93	95	96	97	98	98	99
T6	20	36	49	60	69	75	81	85	89	91	93	95	96	97	98	98	99	99
T7	22	39	53	63	72	79	84	88	91	93	95	96	97	98	99	99	99	99
T8	23	42	56	67	75	81	86	90	93	95	96	97	98	99	99	99	99	99
T9	25	44	59	69	78	84	88	92	94	96	97	98	99	99	99	99	99	99
T10	27	47	61	72	80	86	90	93	95	97	98	98	99	99	99	99	99	99

Across the top is the number of cards of a particular type you have in your deck. Down the left side are turns. Assuming you are on the play (don't draw on turn 1) and draw one card per turn, the chart will tell you the chance to draw at least one of the cards of this type by that turn. You can see the "40" from our Lightning Bolt example in the top row.

What's this have to do with mana bases?

If you built a deck with only eight blue cards and 28 red then you might be tempted to distribute your lands in an even proportion. 8/36 = 22% blue and 28/36 = 78% red. Maybe we should balance our mana base 22/78? 22% of 24 lands is five or six blue lands. Say we round up to to six. How would this perform?

Experienced players know this is a recipe for trouble. A spell in hand without the right mana source is very costly. Dead cards make you lose games. Conversely an off color mana source isn't nearly as costly, even a Chandra's Phoenix can make use of a turn three Island.

Before we go further we need a little more information about this mystery deck. Let's assume its a red aggro deck with 4 Delver of Secrets and 4 Mana Leaks. This is important because now we know we need to cast these spells early, Delver ideally on turn one, and Mana Leak needs to be an option on turn two.

Looking back up at our chart we notice a few interesting points. First, the chance of a Delver in our opening hand is 40%. The chance of a blue card in our opening hand is 65%! By turn three there is 75% percent chance that we'll draw one of our eight blue spells. The big takeaway here is you will draw your splash color spells frequently, so you better have the mana to cast them.

What's the chance of having a blue mana source in our opening hand? With six blue sources we can consult the chart to see. 54%. What does that mean? As you're probably beginning to suspect, our chance of having a Delver on turn one with no blue mana to cast him is pretty high. How high exactly?

Unfortunately, it's not as simple as multiplying .40 * (1 - .54) because the two events are not independent (though this is a good approximation). Long story short, we need an expanded version of the hypergeometric distribution called the multivariate hypergeometric distribution. Sounds fancy. I'll spare you the details. The chance of drawing a Delver on turn one with none of your six blue sources is 20%. This means, if you use this mana base, one in five games your mana base will fail you on turn one.

I wrote some code to calculate the probabilities with a variety of different number of spells and lands. This table shows the chance of a mana base failure (spell but no land) on turn 1, assuming seven cards. Across the top is the number of spells of a particular type. Down the left side is the number of lands (or sources) in your deck.

Single mana base failure for turn 1

	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21	22	23	24
4 lands	08	14	20	25	30	34	38	41	44	46	48	50	52	53	54	55	56	57	57	58	58	59	59	59
5 lands	07	13	18	23	27	30	33	36	39	41	43	44	46	47	48	49	49	50	50	51	51	51	52	52
6 lands	06	11	16	20	24	27	30	32	34	36	38	39	40	41	42	43	43	44	44	44	45	45	45	45
7 lands	05	10	14	18	21	24	26	28	30	32	33	34	35	36	37	37	38	38	39	39	39	39	39	40
8 lands	05	09	12	16	18	21	23	25	26	28	29	30	31	31	32	32	33	33	34	34	34	34	34	34
9 lands	04	08	11	14	16	18	20	22	23	24	25	26	27	27	28	28	29	29	29	29	29	30	30	30
10 lands	04	07	10	12	14	16	18	19	20	21	22	23	23	24	24	24	25	25	25	25	25	26	26	26
11 lands	03	06	08	10	12	14	15	16	17	18	19	20	20	21	21	21	21	22	22	22	22	22	22	22
12 lands	03	05	07	09	11	12	13	14	15	16	16	17	17	18	18	18	18	19	19	19	19	19	19	19
13 lands	02	05	06	08	09	10	11	12	13	14	14	15	15	15	15	16	16	16	16	16	16	16	16	16
14 lands	02	04	06	07	08	09	10	11	11	12	12	12	13	13	13	13	13	14	14	14	14	14	14	14
15 lands	02	03	05	06	07	08	08	09	10	10	10	11	11	11	11	11	11	12	12	12	12	12	12	12
16 lands	02	03	04	05	06	07	07	08	08	09	09	09	09	09	10	10	10	10	10	10	10	10	10	10
17 lands	01	03	04	04	05	06	06	07	07	07	07	08	08	08	08	08	08	08	08	08	08	08	08	08
18 lands	01	02	03	04	04	05	05	06	06	06	06	06	07	07	07	07	07	07	07	07	07	07	07	07
19 lands	01	02	03	03	04	04	04	05	05	05	05	05	06	06	06	06	06	06	06	06	06	06	06	06
20 lands	01	02	02	03	03	03	04	04	04	04	04	05	05	05	05	05	05	05	05	05	05	05	05	05
21 lands	01	01	02	02	03	03	03	03	03	04	04	04	04	04	04	04	04	04	04	04	04	04	04	04
22 lands	01	01	02	02	02	02	03	03	03	03	03	03	03	03	03	03	03	03	03	03	03	03	03	03
23 lands	01	01	01	02	02	02	02	02	02	02	02	03	03	03	03	03	03	03	03	03	03	03	03	03
24 lands	00	01	01	01	01	02	02	02	02	02	02	02	02	02	02	02	02	02	02	02	02	02	02	02

You can use the chart to check our 4 Delvers and 6 sources. You can see that in order to reduce the turn one delver failure to at most ten percent (one in ten games) you need to have at least 11 blue mana sources. If you had eight one drops of a particular color you can see to have at most ten percent failure on turn one you'd need 15 lands.

We can generate this table for any turn. Here's the table for turn two:

Single mana base failure for turn 2

	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21	22	23	24
4 lands	08	15	21	26	31	35	38	41	43	45	47	49	50	51	52	53	53	54	54	54	55	55	55	55
5 lands	07	13	18	23	27	30	33	35	37	39	41	42	43	44	45	45	46	46	46	47	47	47	47	47
6 lands	06	11	16	20	23	26	28	30	32	34	35	36	37	38	38	39	39	39	40	40	40	40	40	40
7 lands	05	10	14	17	20	22	24	26	28	29	30	31	32	32	33	33	33	34	34	34	34	34	34	34
8 lands	05	08	12	15	17	19	21	22	24	25	26	26	27	28	28	28	28	29	29	29	29	29	29	29
9 lands	04	07	10	13	15	16	18	19	20	21	22	22	23	23	24	24	24	24	24	25	25	25	25	25
10 lands	03	06	09	11	13	14	15	16	17	18	19	19	19	20	20	20	20	21	21	21	21	21	21	21
11 lands	03	05	07	09	11	12	13	14	15	15	16	16	16	17	17	17	17	17	17	17	18	18	18	18
12 lands	02	05	06	08	09	10	11	12	12	13	13	14	14	14	14	14	14	15	15	15	15	15	15	15
13 lands	02	04	05	07	08	09	09	10	10	11	11	11	12	12	12	12	12	12	12	12	12	12	12	12
14 lands	02	03	05	06	06	07	08	08	09	09	09	09	10	10	10	10	10	10	10	10	10	10	10	10
15 lands	01	03	04	05	05	06	07	07	07	08	08	08	08	08	08	08	08	08	08	08	08	08	08	08
16 lands	01	02	03	04	05	05	05	06	06	06	06	07	07	07	07	07	07	07	07	07	07	07	07	07
17 lands	01	02	03	03	04	04	04	05	05	05	05	05	05	05	06	06	06	06	06	06	06	06	06	06
18 lands	01	02	02	03	03	03	04	04	04	04	04	04	04	04	05	05	05	05	05	05	05	05	05	05
19 lands	01	01	02	02	03	03	03	03	03	03	04	04	04	04	04	04	04	04	04	04	04	04	04	04
20 lands	01	01	01	02	02	02	02	03	03	03	03	03	03	03	03	03	03	03	03	03	03	03	03	03
21 lands	00	01	01	01	02	02	02	02	02	02	02	02	02	02	02	02	02	02	02	02	02	02	02	02
22 lands	00	01	01	01	01	02	02	02	02	02	02	02	02	02	02	02	02	02	02	02	02	02	02	02
23 lands	00	01	01	01	01	01	01	01	01	01	01	01	01	01	01	02	02	02	02	02	02	02	02	02
24 lands	00	00	01	01	01	01	01	01	01	01	01	01	01	01	01	01	01	01	01	01	01	01	01	01

Here we can see that if we want to reduce turn two mana leak failure to at most ten percent then we need 11 blue sources. However, if we want to reduce any blue failure (delver or mana leak) by on turn two to at most 10 percent then we have to go up to 13 lands (we count all eight blue cards as a type).

Based on this information how should we balance our mana base? Let's say we can use a set of Volcanic Islands (simpifies the turn one math). If we want to stick with 24 lands and we split evenly then we'll have 4 duals, 10 Islands, and 10 Mountains. 14 sources for each color. Looking over the chart you see that this would significantly reduce our chance for blue failure without significantly hurting our chance for red failure. This is really the key point of this article. Evenly split mana bases are more efficient than unbalanced mana bases. Going from 6 to 7 Islands helps you a lot more than going from 18 to 17 Mountains hurts you.

However, what mana base is actually optimal for your deck depends very heavily on the details of your deck. Once thing we haven't considered yet is double colored mana cost. Say you have a Chandra's Phoenix you want to cast on turn three. What's the chance of failure in that case? This time we need to have two red sources.

Double mana base failure for turn 3

	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21	22	23	24
4 lands	14	26	36	44	52	58	64	68	72	75	78	80	82	83	85	86	87	87	88	88	89	89	89	89
5 lands	13	24	34	42	49	55	60	64	68	71	73	75	77	79	80	81	81	82	83	83	83	83	84	84
6 lands	12	23	32	39	46	52	56	60	63	66	69	70	72	73	74	75	76	76	77	77	77	78	78	78
7 lands	11	21	30	37	43	48	52	56	59	61	63	65	66	68	69	69	70	70	71	71	71	71	72	72
8 lands	11	20	27	34	39	44	48	51	54	56	58	60	61	62	63	63	64	64	65	65	65	65	65	65
9 lands	10	18	25	31	36	40	44	47	49	51	53	54	55	56	57	57	58	58	59	59	59	59	59	59
10 lands	09	16	23	28	33	36	40	42	44	46	48	49	50	51	51	52	52	52	53	53	53	53	53	53
11 lands	08	15	21	25	29	33	36	38	40	41	43	44	45	45	46	46	46	47	47	47	47	47	47	47
12 lands	07	13	18	23	26	29	32	34	36	37	38	39	40	40	41	41	41	41	42	42	42	42	42	42
13 lands	06	12	16	20	23	26	28	30	31	33	34	34	35	35	36	36	36	36	37	37	37	37	37	37
14 lands	06	11	15	18	21	23	25	26	28	29	29	30	31	31	31	32	32	32	32	32	32	32	32	32
15 lands	05	09	13	16	18	20	22	23	24	25	26	26	27	27	27	27	28	28	28	28	28	28	28	28
16 lands	04	08	11	14	16	18	19	20	21	22	22	23	23	23	23	24	24	24	24	24	24	24	24	24
17 lands	04	07	10	12	14	15	16	17	18	19	19	19	20	20	20	20	20	20	20	20	20	20	20	20
18 lands	03	06	08	10	12	13	14	15	15	16	16	17	17	17	17	17	17	17	17	17	17	17	17	17
19 lands	03	05	07	09	10	11	12	13	13	13	14	14	14	14	14	14	15	15	15	15	15	15	15	15
20 lands	02	05	06	08	09	09	10	11	11	11	12	12	12	12	12	12	12	12	12	12	12	12	12	12
21 lands	02	04	05	06	07	08	08	09	09	09	10	10	10	10	10	10	10	10	10	10	10	10	10	10
22 lands	02	03	04	05	06	07	07	07	08	08	08	08	08	08	08	08	08	08	08	08	08	08	08	08
23 lands	02	03	04	04	05	05	06	06	06	06	07	07	07	07	07	07	07	07	07	07	07	07	07	07
24 lands	01	02	03	04	04	05	05	05	05	05	05	05	05	05	06	06	06	06	06	06	06	06	06	06

Wow! That's a lot higher failure rate! Doubles are costly. To get at most ten percent failure for a turn three Phoenix we'd need 17 red sources. So what do we do? Well that depends. You'd have to balance the failures to meet your goals. With this build I'd probably and go something like 12/8/4. This would give a pretty even distribution of failure. I'd also wonder if the eight blue cards were worth the trouble.

Here are the complete failure tables. You can use these to help build optimal mana bases for your decks.

Conclusions

We can consult the charts and tweak individual decks, but what can we learn in general?

From the failure tables, if you require 10% or less failure rate on the first turn your spells are available to cast then:

For singles:
To support 4 cards at cmc 1 or less you need 11 lands.
To support 8 cards at cmc 1 or less you need 15 lands.
To support 4 cards at cmc 2 or less you need 11 lands.
To support 8 cards at cmc 2 or less you need 13 lands.
To support 12 cards at cmc 2 or less you need 14 lands.
To support 4 cards at cmc 3 or less you need 10 lands.
To support 8 cards at cmc 3 or less you need 12 lands.
To support 12 cards at cmc 3 or less you need 13 lands.
To support 16 cards at cmc 3 or less you need 13 lands.


Btw 10-15 min for singles. 15 can do anything except excessive one drops. 4 duals and 10 of each is 14 of each.

For doubles:
To support 4 cards at cmc 2 or less you need 20 lands.
To support 8 cards at cmc 2 or less you need 23 lands.
To support 4 cards at cmc 3 or less you need 18 lands.
To support 8 cards at cmc 3 or less you need 21 lands.
To support 4 cards at cmc 4 or less you need 17 lands.
To support 8 cards at cmc 4 or less you need 19 lands.
To support 4 cards at cmc 5 or less you need 16 lands.
To support 8 cards at cmc 5 or less you need 17 lands.
To support 4 cards at cmc 6 or less you need 15 lands.
To support 8 cards at cmc 6 or less you need 16 lands.


Supporting doubles is much more demanding than singles, even if singles are cheaper and more numerous. Double is a commitment.

Eight duals split is 8/8/8, which is 16 per color. This can support some doubles but not most. To be smooth with doubles you have to tilt the mana base. With 4 duals you can go down to 8/12/4 which is 12/16, and this is pushing it. With 8 duals you can go 6/10/8 which is 14/18. This is safe in most regards. This means you generally want to avoid decks with doubles from different colors unless they are expensive or you have a lot of dual sources.

You very rarely want to use less than ten sources.

The 10% failure rate is arbitrary. You can certainly play decks with a higher rate successfully. Finding the balance between a smooth mana base and powerful cards is part of the deck building challenge.

I hope you found this article useful. I learned a lot putting it together. I used to have to build mana bases entirely off a gut feeling. Now it's a slightly more informed gut feeling.