Magic Math: You Should Probably Know These Probabilities

My first disclaimer at the beginning is that, if you run cantrips such as or , it will change a lot of the following numbers. For the purpose of this article, I look mostly at generic draws without cantrips. In addition, let's go over what "There is a 10% chance" actually means. Mathematically, it can be rewritten as 10/100, 1/10, or 0.1. In practice, you can think of it as "If this situation happened ten times, it would work once"; with 25% "If this situation happened four times, it would work once" or "If this situation happened 100 times, it would work 25 times."

The next thing I want to address is the Gambler's Fallacy. If you roll the dice, as you do at the beginning of every match, and you keep losing the die roll, because you get one, two, or three, you might think there is a higher chance that you'll win the next one. Wrong! This is the Gambler's Fallacy. Each of these rolls is an independent event, and one roll does not affect the next. In pure Magic terms, just because you already took two mulligans to find , does not mean that now the chance of finding it in the open hand is higher than it was previously. What it does mean is that with each mulligan you get to reroll the probabilities to see if you hit that ~40% this time.

Last but not least, remember that mathematically getting a six on a die roll five times in a row is just as impressive as getting a sequence of two, five, four, one, and one. It's just that we pay more attention to those sixes and ones, which we perceive as more special.

For my calculations, I used this hypergeometric calculator, which should be easy to use for any Magic player. I principally assumed 60-card decks but also included some of the corresponding numbers for decks accompanied by as they are popular enough to make 80 cards relevant as well. Not least, it allows us to see how much the probabilities change when you add twenty cards.

Probabilities for Opening Hands

• Drawing at least one card of which there are four copies in the deck:
  39.9% (31.2% for Yorion decks)

It's about ten percentage points off from a coin flip. In other words, you're more likely not to have it than you are to have it. Still, in roughly four out of ten games, you can expect that to show up in the first seven.

• Drawing at least one card of which there are eight copies in the deck:
  65.4% (53.6% for Yorion decks)

This stat is useful for all the Bogles players out there who want a copy of either or on turn one. You can also use it in Burn where you want to start every game with a or , or for any number of decks that run and . It's worth pointing out that it's not twice as high as the probability with a single playset of cards in the deck.

• Drawing one or more cards of which there are three copies in the deck:
  31.5% (24.3% for Yorion decks)

Sometimes you want to know how likely you are to open on a card that you need but actually never want to draw. In a cascade deck that runs three copies of or , you'll find one in the opener roughly a third of the time—at least at sixty cards. When you want to see a card less often, Yorion builds are at an obvious advantage.

• Drawing at least one copy of a sideboard card when you add X copies …
  One copy: 11.7% (Yorion: 8.75%)
  Two copies: 22.1% (Yorion: 16.8%)
  Three copies: 31.5% (Yorion: 24.3%)
  Four copies: 39.9% (Yorion: 31.2%)

These stats might help you assess better how many copies of a sideboard card you actually need. Bear in mind that those are the numbers for the opening hand alone, which is most relevant with for example . If you need a sideboard card on the second turn, you get two more draws which changes the math.

• Drawing exactly three lands with 24 total in your 60-card control deck:
  30.9%.

In about three out of ten times you'll get the desired three-land hand. If you want at least three lands but also accept, say, five, then the probability goes up to 58.8%.

• Drawing exactly two lands with twenty total in your 60-card aggro deck:
  32.4%.

When you play decks like Burn, you want to see three lands maximum for the entire game. The probability that by turn four you will have drawn exactly three lands is 28.2% on the play and 25.6% on the draw.

Probabilities Including Further Draws

• Drawing a second land when you kept a hand with one, on the draw (OTP) and on the play (OTP) …
  Deck with 19 lands: 34.5% (OTP), 56.8% (OTD)
  Deck with 20 lands: 35.8% (OTP), 59.3% (OTD)
  Deck with 21 lands: 37.7% (OTP), 61.7% (OTD)
  Deck with 22 lands: 39.6% (OTP), 64.0% (OTD)
  Deck with 23 lands: 41.5% (OTP), 66.3% (OTD)
  Deck with 24 lands: 43.4% (OTP), 68.4% (OTD)
  Deck with 25 lands: 45.3% (OTP), 70.5% (OTD)

As we can see, the probability on the draw is much higher. You can see 20–30 percentage points difference between the two situations.

• Finding a second land when you kept a one-lander with in a 19-land deck:
  56.8% (OTP), 72.1% (OTD).

This crucially pertains to one of the most played Modern decks in Blue-Red Murktide. It runs 19 lands and cantrips. As we can compare it with the data from the previous point, we see the probability goes way up with cantrips in the mix.

• Drawing at least one cascade spell by turn three:
  75.1% (OTP), 79% (OTD).

Some decks need a specific effect, but you don't need it in the opener. As is the case in cascade decks, you need to have drawn at least one of either or by turn three. In case of , there are a lot of cyclers that change the math. Let's assume you've cycled twice by turn three. Then the probabilities will look as follows: 82.4% (OTP), 85.3% (OTD). To hit the 90% mark, you'd need to have cycled five (OTP) or four (OTD) times by turn three to draw at least one cascade spell. It's technically doable with the help of and looking at two cards.

The Relevance of Deck Thinning

Often people will debate whether it's worth it to thin out the deck with fetch lands such as . When you fetch, you remove one card from the deck. If you wanted to draw a piece of removal and you have six in a deck that's now 50 cards big, you'd have 12%. Supposing you'd fetched before that draw step, so now you're drawing from a library of 49 cards instead of 50, the chance would go up to … 12.2%. While technically more, the popular argument is that 1 life lost, which is 5% of the full life total, is way more relevant than an increase of a fraction of a percent.

However, if your deck is already slim, say with twenty cards in it, the relevance of thinning increases. If we take the above example, you'd have 30% to draw the removal spell before the fetch and 31.6% after. That is something to consider during gameplay. But when you build a monocolored deck without graveyard/landfall shenanigans, where fetch lands would serve no other purpose than thinning, you're better off with 5% more life. Probably.

I hope you've found this breakdown useful. I certainly had a lot of fun delving into the numbers of our beloved game. As always, hold my hand, and let's pass the turn together. Cheers!

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