When I add a card to my deck, I often wonder: what is the chance I’m going to draw it? Maybe it’s a mythic bomb, and I’m super excited to cast it. Maybe
its game two and I’m adding an answer. What is the chance I’ll draw my answer? Should I add two? How much better is my chance if I add two?

My Awesome Bomb

Let’s start simple. If I have a single copy of a bomb in my deck, what is the chance I’ll get to use it to crush my opponent?

To be concrete, I’ll say that I need to have the card by turn nine for it to have a real impact on the game. Is that true? It depends a lot on the
specifics of the card, but turn nine seems reasonable for most bombs, cards that can win the game on their own, but don’t immediately stabilize me if I’m
behind. Think Niv-Mizzet, Dracogenius or Jace, Architect of Thought.

Assuming I’m on the draw, I will have drawn 16 cards by turn nine: 7 in my starting hand and 9 during turns. Given a 40-card deck, the chance of one of
those 16 being the single copy of my bomb is 16 over 40, or 40%.

40% is a two in five chance. I can expect to play my bomb in about two of every five games. That’s a little more often than once in a three-game match.

Adding Answers

What if I’ve got more than one copy of a card in my deck? What if I’ve got two or three? Maybe I’m siding in two copies ofSundering Growth to deal with my opponent’s Collective Blessing. What is the chance I will draw one or more of them by turn
nine? That calculation is slightly more complicated. Here are the results.

The probability doesn’t double or triple, as you might naively expect, but the chance does go up dramatically. With two answers in my deck, I have about a
65% chance of getting to at least one in a timely fashion.

Holding Out For a Better Answer

Sometimes in the middle of a game I find myself pondering the probability that I will draw a particular card. Usually I’m making some critical play
decision and the correct answer depends heavily on what my next few draws will be. For example, take the following situation. I’m Golgari with a red splash
for Dreadbore. My opponent is pecking away with aVassal Soul. I have theDreadbore in hand and the mana to cast it, but my deck also has twoTowering Indriks. Do I waste the Dreadbore on a stupid 2/2 or do I hold out, hoping to draw one of my blockers?
Knowing the correct play depends on knowing the probabilities involved. Let’s explore those probabilities.

Assuming it’s the start of my sixth turn and my deck has one copy of the card I’m hoping for, the chance of me topdecking that card is 3.6%. (That comes
from the fact that there are 28 cards left in my library at the start of turn six, one of which is the card I want, so the chance is 1 in 28.) Those are
abysmal odds. Maybe I should just use Dreadbore. But before getting too
depressed, let me look at a few other factors that affect the probability.

If I have two or three copies of the card left in my library, the chance increases. The probability doubles or triples, respectively.

Similarly, if I give myself more than a single turn to draw the card, the chance goes up. Again the probability increases arithmetically. (This table
assumes I have a single instance of the card in my deck.)

Finally I will consider both of those factors simultaneously. Here is a table of the probability for all cases.

The chances are getting well above 4% and into the 20% to 30% range. Now I am at probabilities worthy of consideration when making play decisions.

The Life Cost

Let’s return to the original situation. I’m being attacked by a Vassal Soul in
the air. Let’s say it’s the middle of my opponent’s attack phase when I’m making the decision about whether to use Dreadbore on my turn. Let’s say I have ten life. With ten life I have four
draws before I have no choice but to cast Dreadbore on the 2/2 flyer to avoid
death. If I have two copies of Towering Indrik in my deck, the previous table
tells me that I have a 27% chance of drawing one in the four turns before I’m forced to cast Dreadbore. That’s about a one in four chance, which seems like it might be a
chance worth taking.

But what cost am I paying for this chance to conserve Dreadbore? If I never
draw a Towering Indrik and am forced to cast Dreadbore, the cost will be eight life. The cost is less if I draw my blocker
earlier. The earlier I draw Towering Indrik, the cheaper the life cost. The
lowest possible cost is two life, since I’m already in my opponent’s attack phase and the flyer is attacking.

How much life can I expect to pay, on average, for the chance to conserveDreadbore? I’ve used the Magic Probability Toolkit to calculate the average amount of life I lose before I
either draw a blocker or am forced to cast Dreadbore. Here are the results.

With two Towering Indriks in my deck, I can expect to pay about seven life for
a one in four chance of conserving Dreadbore.

Is that worth it? That depends. How much other removal do I have in my deck? How many cards are left in my opponent’s hand? Do I know of some brutal bomb
in my opponent’s deck that Dreadbore is my only answer for? Are we racing? Will
killing the Vassal Soul immediately put me ahead in that race? These and a
million other considerations need to be taken into account when making the decision. But the probability of saving Dreadbore is definitely one of the important considerations, and now I have a
sense for that probability.

Later in the Game

So far I’ve assumed that the Vassal Soul situation came up on the attack phase
before my sixth turn. But the probabilities change depending on which turn it is. The longer the game has gone without me drawing my answers, the more
likely it is that I’ll draw them now, simply because there are fewer cards in my library.

I’ve repeated the same calculation as before, but this time I’ve varied the turn of my opponent’s initially attack. I did the calculation only for the case
where I have two copies of Towering Indrik in my deck. Here are the results.

As expected the probability goes up later in the game. But the change is not very dramatic, only a few percentage points, even as late as turn 12.
Similarly, the change in the average life lost is minimal. There is clearly not a strong need to account for which turn it is when making this play
decision.

Takeaways

If you have one copy of a card in your deck, you’ll get to play that card in 40% of your games. If you have two copies, you get to play it 65% of the time.

During the mid game, if you really need to draw a particular card, there’s a 14% chance you will draw it within four turns. If you have two copies in your
deck, that chance rises to 27%. Three copies raise the chance to 38%.

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source code for the Magic Probability Toolkit