Deckbuilder's riddle
Standard forum
Posted on March 4, 2013, 2:25 p.m. by hawgdriver
Which deck has a higher probability of having 4 lands out on turn 4, ready to cast a 4 mana spell:
24 Swamps and 4 Sign in Blood, or
28 Swamps?
Next riddle:
Assume you chose the deck with a higher probability of having 4 Swamps out on turn 4. What is that probability? That is, if you played 100 games, no mulligans, played Sign in Blood on turns 2 and 3 if one or more copies were in your hand, how many of those games would you be able to cast a 4 mana spell on turn 4?
TorrentFreak says... #3
A deck with 24 Swamp and 4 Sign in Blood would have the higher probability of hitting 4 lands by turn 4. My reasoning is that the extra two draws you get from Sign in Blood gives the deck more chance of hitting land. 28 Swamp has a high % chance of hitting land in the opening hand, then that % drops off.
not sure about the probability each game, but I am guessing 65% of the time, or 65/100 games
March 4, 2013 2:47 p.m.
Depends a bit on whether or not you are on the play or the draw...
March 4, 2013 3:02 p.m.
Without Sign in Blood... the only way of this is:
T1: - Swamp
T2: Draw - Swamp
T3: Draw - Swamp
T4: Draw - Swamp
With Sign in Blood...
T1: - Swamp
T2: - Draw - Swamp - Possibly a Sign in Blood (Draw 2)
T3: - Draw - Swamp (swamp could be after the SiB) - Possibly a Sign in Blood (Draw 2)
T4: - Draw - Swamp
So, you need to calculate odds of getting 4 swamp in the first 10 cards in a 28 land deck of 60 cards, vs.
4 swamps in the first 10 cards of a 24 land 60 card deck when no Sign in blood is drawn + 4 swamps in the first 12 cards of a 24 land 60 card deck when 1 Sign in blood is drawn + 4 swamps in the first 14 cards of a 24 land 60 card deck when 2 Sign in blood is drawn
Thus you also need to know the chances of getting Sign in blood by turn 2, or 3, plus if you need to know if you get one by turn 2, what your odds are of having one by turn 3...
It gets complex... and I'm just going to leave it at that for now.
March 4, 2013 4:50 p.m.
hawgdriver says... #10
FYI I ran a simulation using 16,400+ runs for this problem.
March 4, 2013 4:58 p.m.
hawgdriver says... #12
Any spreadsheet and the properly constructed set of formulas applied to 16,000+ sets of 60 random numbers. I used MS Excel.
March 4, 2013 5:09 p.m.
hawgdriver says... #13
I'll post the answer to this riddle tomorrow afternoon.
March 4, 2013 5:23 p.m.
I detest the application of the term 'riddle' to mathematical formulae.
March 4, 2013 5:34 p.m.
Agreed. This is a math problem. Nothing more. A pretty easy one too I believe.
March 4, 2013 6:16 p.m.
hawgdriver says... #16
From Merriam-Webster, available here
Definition of RIDDLE
1: a mystifying, misleading, or puzzling question posed as a problem to be solved or guessed : conundrum, enigma
Rayenous, I am in disquiet knowing that you are in a state of detestation! Please test soon!
Squire 1, please share your answer to this easy non-riddling math problem!
:D
Lighten up guys. (if you want, no pressure from me -- :P )
March 4, 2013 6:37 p.m.
hawgdriver says... #17
The unexpected and somewhat mystifying result is that it is preferable to run more swamps than Sign in Blood. In this case, there is a 78% chance of having 4 untapped lands after the turn 4 land drop (assuming 60 cards, playing first) when running 28 swamps. With 24 swamps and 4 Sign in Blood, there is a 72% chance of the same.
This is the most elegant way I can think of to understand why this is true. When running 24 swamps in a 60 card deck, the expected land value of any given card is 0.4.
Say you draw a hand with a Sign in Blood. What is the expected number of swamps that card will provide? Since it is the equivalent of two cards, it will provide 0.4 swamps with the first card, and 0.4 swamps with the second card -- that is, it is worth 0.8 swamps.
So, even though the Sign in Blood reaches deeper into the deck, there is a greater likelihood of casting bigger spells by simply using swamps. The chart below shows the probabilities of having 4 lands available on turn 4 for the range of possible deck configurations using the same number of cards to provide mana sources (the highest line is the line with 0 Sign in Blood--running just swamps, the lowest line shows the effect of replacing swamps with Sign in Blood).
hawgdriver says... #2
To be clear, the swamps must on the board, untapped, on turn 4.
March 4, 2013 2:32 p.m.