Should I mill myself if I have graveyard synergies in my deck, but haven't drawn them yet?

Question

It's well-established that one should usually choose to loot if one can. It's the strictly superior play, and there are only a few valid reasons not to do so.

This question is about a different scenario. If I have cards that use the graveyard in my deck, but have not drawn them yet, under what circumstances should I mill myself? Concrete example:

My board: 20 life, no cards in hand, no cards in graveyard, 40 cards in deck, lots of lands, a Hill Giant, Skull Prophet.

Opponent's board: 20 life, no cards in hand, lots of lands, two Grizzly Bears.

Right now neither of us can attack. If I do, he double blocks and I trade down. If he attacks I simply eat his creature. The board is stalled.

It's the end of my opponent's turn. I have Survivors' Bond still in my deck, but I haven't drawn it yet. Should I activate Skull Prophet?

I'm wondering if the answer is "yes" (because I can potentially mill good cards for Survivors' Bond), or "no" (because I can potentially mill the Survivors' Bond itself). If the answer is "it depends", then I'd like to know under what circumstances it becomes desirable - e.g. how many copies of Survivors' Bond vs. good creatures do I need in my deck before it's preferable to mill myself?

Edit: Since this question became a HNQ, here's a nontechnical version.

There are two zones for cards, "in the deck", and "in the graveyard". I have a deck of X cards, from which I draw a card every turn. There are Y copies of a card ("Survivors' Bond") in the deck that prefer cards to be "in the graveyard". Every turn I'm given the option to put two cards from my deck into the graveyard. Should I do it because it makes my Survivors' Bond better, or should I not do it because I could end up putting Survivors' Bond into the graveyard? If it depends on what X and Y are, for what parameter range does it become preferable?

Answer 1

Let's assume the game will end within 13 turns. In those 13 turns, you will draw 13 cards, and since you have no information where the Survivors' Bonds are, the milling has no effect on how fast you draw one or how likely you are to draw one. If you don't mill at all, you will draw the top 13 cards; if you mill every turn, you'll draw the cards at positions divisible by three. (Well, if you're unlucky enough to mill all of them, you can stop milling of course.)

Milling does however increase the value of a Survivors' Bond; without creatures, it's a dead card, and the more options you have, the better.

So bottom line: always activate Skull Prophet.

Answer 2

Yes. If you have no information about the order of cards in your deck, then you should practically always mill yourself.

Milling yourself is your strategy. It's the main purpose of your Skull Prophet. Ignoring the other 3 Bonds for the moment, and absent any information about the order of cards in your library, your Survivors' Bond has a 2/40 (5%) chance to be in the top 2 cards and be wasted by the mill, and a 38/40 (95%) to get closer to the top spot instead.

Only when those percentages flip around, i.e. the chance to mill them away becomes bigger than the chance of moving them up, should you stop milling. But that only happens when you have only very few cards left in the library, usually not a concern in even remotely competitive games.

Of course, the counterargument would be that if you mill 2 cards for every card you draw, you can expect to lose 2-3 Bonds to milling each game. To ensure that you're going to draw a key card that you need in your hand rather than your graveyard is a matter for deck construction, either by including multiple similar cards (or tutors), or more generic card draw.

Answer 3

Milling:

• doesn't make drawing a card more or less likely.

• could add potentially good usable cards to your graveyard.

• won't feel good if you mill the card you want to draw, and this is fairly likely to happen (especially if you're milling more cards than you're drawing). It might feel good if you manage to draw it after a lot of milling, but probably not as bad as you'd feel if you accidentally mill it.

• is a significantly worse idea if you have any sort of deck searching or scry (that you haven't used yet), because you wouldn't be able to search cards you've milled. But I assume this doesn't apply.

• is not a great idea if there's a risk of running out of cards in your library. I assume this also doesn't apply.

So milling is good, from a strictly mathematical point of view.

But psychologically it's not great. This probably shouldn't be that important, but it's worth keeping in mind.

Why doesn't milling make drawing a card more or less likely?

Intuitively:

You can consider milling to be equivalent to putting those cards at the bottom of your library. Either way, you won't draw those cards and the order of the rest of the cards are the same. Since you have no knowledge of the order of the cards, reordering your library in this way can be considered equivalent to simply shuffling your library. Shuffling your library doesn't make it more likely to draw any given card (if you have no information about the order of the cards).

More concretely:

To determine the probability of drawing a specific card: Assuming the cards in your library are in a random order, you can just look at the amount of cards you're drawing. Then ask: out of the N cards you currently have in your library, what's the probability of getting the card you want in K cards you draw? You're drawing K out of N cards regardless of whether or not you mill yourself. So the milling is irrelevant.

The probabilities can change as you mill yourself, but this would only be due to information gain. By this I mean knowing what cards you can no longer draw makes drawing any of the other cards more likely. But the information gained would cancel out with the chance of gaining that information. If you didn't know what you were milling (say you remove the top card of your library face-down), it wouldn't change the probabilities. From a probability point of view, this would be exactly equivalent to removing the bottom card of your library (face-up or face-down, respectively).

If you have 1 copy of the card in a deck of 40 cards, the maths for 1 draw goes as follows:

Chance of getting the card before mill = 1/40
Chance of getting the card after mill
    = chance of milling the card * chance of drawing it from deck without card
      + chance of not milling the card * chance of drawing it from smaller deck
    = 1/40 * 0/39 + 39/40 * 1/39
    = 0 + 1/40
    = 1/40
    = Chance of getting a card before mill

More generally, for K copies of the card in a deck of N cards, you'd have:

Chance of getting a card before mill = K/N
Chance of getting a card after mill
    = chance of milling the card * chance of drawing it from deck with 1 fewer copy
      + chance of not milling the card * chance of drawing it from smaller deck
    = K/N * (K-1)/(N-1) + (N-K)/N * K/(N-1)
    = K(K-1)/N(N-1) + K(N-K)/N(N-1)
    = (K(K-1) + K(N-K))/N(N-1)
    = K(K-1 + N-K)/N(N-1)
    = K(N-1)/N(N-1)
    = K/N
    = Chance of getting a card before mill