A common strategy in Magic the Gathering and other trading card games is deck thinning, moving lands from the deck onto the battlefield early in the game. In some ways, deck thinning is troll math. In others, it’s an optimal strategy.
Example: A player taps Bloodstained Mire to use its ability Sacrifice Bloodstained Mire: Pay 1 life, discard Bloodstained Mire into the graveyard, and search the deck for either a Swamp or Mountain, put the land into play, and shuffle the deck. This card lends itself to Red-Black decks as it gives players the option of searching for Swamps or Mountains.
Players maintain that Bloodstained Mire and other deck thinning cards in dramatically alters the probability of drawing certain cards. Lands are removed from the deck and played early in the game, and non-lands are abundant later in the game. A 1/15 chance of drawing a desired card increases to 1/14.
However, 1/15 is only for a hypothetical deck in which every card exists as four multiples. 1/14 is only if you a lucky enough to remove four of the same exact card (unlikely), and that won’t happen with a single deck thinning card.
The probabilistic benefits of deck thinning are next to nil, but spending a few mana now to deploy lands needed later is worthwhile. However, deck thinning cards are only useful if a player can play them; while in the deck, or in the hand but without sufficient mana, they merely take the place of an otherwise useful card.
Deck thinning cards are just as likely to be drawn early in the game as they are to be drawn late in the game. If a deck thinning card is drawn early in the game, a land would have been as profitable as the thinning card (or more profitable considering the price of activating Bloodstained Mire). If a deck thinning card is drawn late in the game, it’s a complete waste; sorceries, instants, and creatures would be better.
Deck thinning can be a false economy. A 1/14 chance of drawing a certain card is hardly an improvement over 1/15. Bloodstained Mire is particularly undesirable, as it requires draining a life point and sacrificing itself. It also must be tapped to be activated, and thus is affected by land tapping spells such as Manabarbs.
A good way to change the odds of drawing certain cards is to have more or less of them in the deck. Instead of Bloodstained Mire, use Badlands. Badlands directly serves as a swamp or mountain land as the player wishes, does not drain health, does not have to be discarded after use, and does not shuffle the deck.
Scrying, fetching, and other draw cards are highly useful in their own right, just not for deck thinning / probabilistic reasons. A card that removes some from the deck and places them on the board, a card that puts two or more cards in the hand, a card that lets players search for any card in the deck, and even a card that puts cards in the graveyard is useful, recursion permitting.
Update: This post was originally titled “Deck Thinning: A False Economy”. It has been revised to acknowledge the direct benefits of fetch cards while still refuting the main of deck thinning theory: that probabilities are drastically altered, and that this is the primary benefit of deck thinning.
Update: Garett Johnson confirms by Monte Carlo simulation that deck thinning is next to worthless.