I have been crunching numbers for the last few days and I wanted to show you some intermediate results. Here's the list of assumptions:
- A deck contains one commander and 99 cards. This calculation ignores partners for now.
- A hand or a mulligan is always 7 cards (London mulligan) where you put n-1 cards to the bottom of the library after the first free mulligan where n is the number of mulligans you've taken.
- The calculations do not take into account any additional draws or draw spells. This is a weakness but we justify this by the fact that you need to choose to keep the hand or mulligan before you see the first draw of the game. Hence you cannot rely on that draw being a land or a ramp spell, for example. Additionally early turn draw spells are scarce, usually don't draw many cards or do not draw cards unconditionally.
- Lands can be any lands although if you use this chart to find the mana available on a specific turn you'll want to assume your final land enters untapped.
- A ramp spell has the mana value 0, 1, or 2. Not exactly relevant for the calculations themselves but helps explain why we defined a keepable hand the way we did.
- A keepable hand consists of one of the following combinations: 2 lands and 1 ramp spell, 2 lands and 2 ramp spells, 3 lands, or 3 lands and 1 ramp spell. We discarded the 4 land hand as an opening hand for two reasons: it skews the results towards lands way too much and I bet keeping a 4 land hand is not ideal if you need to mulligan to 5 cards. Similarly this calculation does not consider the possibility of a single land that has for example a Sol Ring and a mana value 2 ramp spell (for example Arcane Signet) because the math would be really complicated and such a hand is still a risky hand.
- We take 3 mulligans maximum (i.e. you see four hands: opening hand, the free mulligan and two regular mulligans) because 5 cards is still usually a keepable hand, a nice mixture of lands, ramp spells and other cards. Subsequent mulligans are somewhat rare.
The number of lands is in the leftmost column and the number of ramp spells is the top row. The value in the cell is the probability of drawing a keepable hand within three mulligans (down to 5 cards).
How to read the chart:
- Example 1: if your deck has 35 lands and 9 ramp spells you take "35" from the lands column and follow that horizontally until you get to "9" on the top row. The value in the cell reads 88.2% which is your probability of finding a keepable hand within three mulligans.
- Example 2: your deck has 38 lands and you want to find the ideal number of ramp spells in order to maximise your chance of a good opening hand in three mulligans. You take "38" from the land column and follow that horizontally until you find the cell with the highest value on that row. Now look up the number of ramp spells from the top row. There are two equal probabilities: 88.9% - one at 11 ramp spells and the adjacent at 12 ramp spells.
- Example 3: you have no idea what you're doing. You just want to know what is the ideal composition of lands and ramp. First find the highest value in the chart (for your convenience I've highlighted it in orange) and then look up the number of lands from the left and number of ramp spells from the top. The highest probability of a good hand in the chart is 89.1% at 36 lands and 12 ramp spells.
- Example 4: you're happy if you get screwed in 12.5% of the games (one in eight) you play and you want to know what is the bare minimum number of lands and ramp you should have. You should assume that the number of ramp spells is the more important factor here because that propels you ahead of curve. Lands would not. You prioritise the number of ramp spells and try to find a cell with a chance of 100-12.5=87.5 with the highest number of ramp spells and minimum number of lands. There's a pretty good match at 31 lands and 16 ramp spells (with a probability of 87.6%) so you pick that.
There are lots of rules of thumb for these kind of calculations. I've seen people say "start with 40 lands and subtract two for each ramp spell you have" or "have a total of 50 mana sources in the deck" and so forth. None of these are really true but they're somewhat good approximations for small deviations from 36 lands and 12 ramp spells which was the ideal composition.
Some additional math that is not shown in the chart: if your commander is mana value 4 and you want to maximise the chance of having exactly 1 ramp spell in your hand (that would put you to 4 mana on turn 3) your best bet is to run 14 ramp spells. Following that thought your best number of lands is then going to be 35 if we read the chart.
More math not on the chart: if you think 4 lands is a good opening hand you should optimally play 0 ramp spells and 49 lands. This probably proves that a 4 land hand is not a keepable hand unless it's an emergency and you've had to mulligan way down. Or at least it's not worth optimising for anyway.
How is it all done? Here's the math bit. Normally figuring out a probability like this uses something called a hypergeometric distribution.
Here's a calculator you can use. It's a mathematical tool that takes in (in MtG terms) the number of cards in the deck, the size of your sample, e.g. opening hand (you can plug in any number of draws you want), how many of that card is in your deck and finally how many cards of that kind you want in your sample. This tool always assumes that your deck is divided into two mutually exclusive groups: cards of type A (e.g. lands) and cards of type B (e.g. nonlands). You cannot have "lands, ramp and the rest" as your three categories because this tool does not understand three categories.
The trick is something called "multivariate hypergeometric distribution" where you can plug in many variables. As far as I can tell there aren't any good online calculators for this. However there is a formula that we can use: in short we take the probability of finding lands and multiply that by the probability of also finding ramp and the probability of also finding other cards and divide the entire thing by the total number of possible hands. This formula takes into account the cases (hands) where all conditions are met. Our chart takes into account the chance of drawing 2 lands and 1 ramp spell, 2 lands and 2 ramp spells, 3 lands, or 3 lands and 1 ramp spell for a total of 4 categories. Each category consists of multiplication (X lands
and Y ramp spells
and Z other spells) because we want to find hands that fulfill the criteria but we sum up the chance of each category together because we're fine with any one of the categories. This way we get the total chance of finding a good hand.
For the mulligan part the math is surprisingly easy. We take the chance of
not finding a good hand. For the opening hand it's just 100-(the chance of finding a good hand). For the first mulligan we assume that we fail the opening hand and the mulligan, too. This means we take the chance of
not finding a good hand and multiply it by itself since each mulligan is always 7 cards and we shuffle the deck in between so the two events are independent of each other. If we subtract that chance from 100 we get the chance of finding a good hand in either the opening hand or the mulligan. Thank the London mulligan for being so simple.
Questions? Thoughts? Did I make a mistake? What kind of things would you like me to figure out next? Anything is possible. I'm planning on making this into a primer or get it pinned as a guide so any feedback is welcome and appreciated.
Credits to vaasa as well for helping me figure out a good portion of the math.
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