I have a Magic the Gathering deck where an ideal starting hand can potentially let me win on turn 3. What is the probability of drawing an ideal hand? How is this computed, and what is a general algorithm that can be used to compute ideal hand probabilities for an arbitrary deck and ideal hand?
This is really a probability question, no knowledge of Magic required. All you need to know is the deck is 60 cards with some duplicates, and the starting hand size is 7 cards.
- 10 Forest
- 6 Swamp
- 4 Terramorphic Expanse
- 4 Llanowar Elves
- 4 Elves of Deep Shadow
- 4 Rites of Spring
- 4 Cultivate
- 4 Blood Speaker
- 4 Howling Wolf
- 4 Skyshroud Sentinel
- 4 Wall of Blossoms
- 2 Infernal Spawn of Evil
- 2 Infernal Spawn of Infernal Spawn of Evil
- 4 Infernius Spawnington III, Esq. (for reference, this is a silver-bordered card so not legal in any competitive format)
- 1 Forest
- 1 Llanowar Elves -OR- Elves of Deep Shadow
- 1 Rites of Spring
- 1 Infernius Spawnington III, Esq.
- 3 (anything)
If you are a Magic player, this is how the ideal game goes:
- Turn 1: Play a Forest and Llanowar Elves.
- Turn 2: Play Rites of Spring, discard 4 cards that are not Infernius Spawnington, search for four lands including a Swamp, play the Swamp, play Infernius, say "I'm here!" to do 3 damage, attack for 9 damage.
- Turn 3: Attack again.
Bonus points Although my original question was just asking about the probability of drawing the ideal starting hand, as @Phillip Kendall pointed out, there are some additional details that could increase the probability of accomplishing the desired effect of a 3rd turn win.
- Because the first player does not draw on their first turn, if you go second, you effectively have a starting hand of 8 cards. (My first thought to account for this, is to run the same computation with a starting hand of 7 cards or 8 cards, and average the result.)
- The Forest and Elf must be in your starting hand (or drawn on your first turn), but the Rites or Infernius could be drawn on your second turn. (I am not sure where to start to account for this.)
- If you are going second, and your starting hand is not ideal, you can mulligan in hopes of getting an ideal hand, and still have enough cards to discard 4 with Rites of Spring, and achieve the 3rd turn win. (I am not sure how to account for this.)