# non-intuitive results in random sampling

I play a card game (Magic The Gathering) that involves creating a deck of cards that can loosely be split into two categories: land and spells. You are permitted to play only one land per turn, and the land are used to cast your spells. Due to the one-per-turn limitation, it is advantageous to have access to exactly X land on turn X (otherwise your hand is wasted on land you can't play.)

For purposes of my current experiment, I consider myself to be in a good position if on turn X, I have access to (that is, have either drawn or played) either X, X+1, or X+2 land. Having access to X-1 land means I'm not playing spells of cost X on turn X, while my opponent is, so I'm probably at a disadvantage. Having X+3 or more land on turn X means I don't have a good choice of spells to play. I desire the happy medium, with the potential buffer of 2 extras.

You start the game with 7 cards, and draw an additional card every turn. There are other effects that can cause you to draw more, discard more, play additional lands, or play more powerful lands, but these are edge cases that are not included in this experiment.

In order to gather data, I wrote a small java application that enables me to simulate drawing cards from my deck. The experiment consists of a 40 card deck with either 16 or 17 land. (A good rule of thumb is 40% land.) For simulation purposes, I do monte-carlo of 1 million hand draws.

Here are the results for having 16 land. This table reads "On turn 4, I have access to 5 land 21.9% of the time." The "good" column is the sum of X, X+1, and X+2 for any given turn.

16 Land   Turn
Lands     1      2      3      4      5      6      7      8
0        1.9%   0.9%   0.5%   0.2%   0.1%   0.0%   0.0%   0.0%
1       11.5%   7.2%   4.3%   2.5%   1.4%   0.7%   0.4%   0.2%
2       27.4%  21.0%  15.2%  10.5%   6.7%   4.2%   2.5%   1.4%
3       31.9%  31.0%  27.6%  22.9%  17.8%  13.1%   9.2%   6.0%
4       19.7%  25.2%  28.3%  28.9%  27.3%  24.0%  19.8%  15.4%
5        6.4%  11.5%  17.0%  21.9%  25.4%  27.1%  26.7%  24.6%
6        1.0%   2.9%   5.9%  10.0%  14.7%  19.3%  23.0%  25.4%
7        0.1%   0.4%   1.2%   2.7%   5.3%   8.7%  12.8%  17.0%
8                      0.1%   0.4%   1.1%   2.4%   4.6%   7.4%
9                                    0.1%   0.4%   1.0%   2.1%
10                                                 0.1%   0.4%

Good:   71%    77%    73%    61%    45.4%  30.4%  18.4%   9.9%


In the same manner, here is the result for 17 land.

Lands   1      2      3      4      5      6      7      8
0      1.3%   0.6%   0.3%   0.1%   0.1%   0.4%   0.0%   0.0%
1      9.2%   5.4%   3.0%   1.6%   0.8%   2.8%   0.2%   0.1%
2     24.6%  17.8%  12.2%   7.9%   4.8%   9.9%   1.5%   0.8%
3     32.3%  29.8%  25.1%  19.7%  14.4%  20.8%   6.4%   4.0%
4     22.5%  27.4%  29.4%  28.3%  25.2%  27.2%  16.2%  11.7%
5      8.4%  14.3%  20.0%  24.5%  27.1%  22.4%  25.3%  21.8%
6      1.5%   4.1%   8.0%  12.9%  18.0%  11.7%  25.2%  26.2%
7      0.1%   0.6%   1.8%   4.1%   7.5%   3.9%  16.3%  20.6%
8                    0.2%   0.7%   1.9%   0.8%   6.8%  10.6%
9                           0.1%   0.3%   0.1%   1.8%   3.5%
10                                               0.3%   0.7%
11                                                      0.1%
Good: 66.1%  75.0%  74.5%  65.7%  52.6%  16.4%  24.9%  14.8%


Okay, now that's a lot of data. I've included it for your review.

Here's what I don't get - let's take a look at the "good" results for 16 versus 17, paying very close attention to turn 6 for 17 land.

         Turn 1  Turn 2  Turn 3  Turn 4  Turn 5  Turn 6  Turn 7  Turn 8
16 Land  70.8%   77.2%   72.9%   60.8%   45.4%   30.4%   18.4%    9.9%
17 Land  66.1%   75.0%   74.5%   65.7%   52.6%   16.4%   24.9%   14.8%


It would appear that on turn 7, my liklihood of having 7,8, or 9 land is higher (substantially) than having 6,7, or 8 land on turn 6. How is that value so low on turn 6? I've summed up all the columns, and they of course add up to 100% (barring a couple rounding errors). I've re-ran the simulation multiple times and it consistently comes up with the same value.

Here are the questions that I feel will help me solve this mystery:

• What could possibly cause the "good curve" for 17-land to dip so much on turn 6?
• How could it be more likely that a 16 land deck could have a higher chance of having the same amount of land access than a 17 land deck after the same number of draws?
• Are there formulas that allow me to calculate the "land-likelihood" that might shed insight into this anomaly?
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Just play brainstorm and fetchlands. Then you won't have this problem. – Euler....IS_ALIVE Aug 20 '12 at 17:37
This is for draft simulating :-) – corsiKa Aug 20 '12 at 18:03

How sure are you that your java application is working properly? The whole $6$ lands column is out of line with the rest of the $17$ lands table. It would fit very nicely if it were dropped down one row, as would the resulting Good total of $38.0$%. It really looks as if the entries in that one column are all one row too high.