The Extra Card Fallacy

There seems to be this prevalent idea within the Magic community that you should never run 41 or 61 cards. The general claim is that running the extra card reduces your chances of drawing specific, important cards in your deck, and therefore is bad for your overall results. I intend to show that this claim, while it has some merit, is misleading. The fact of the matter is that there is a very good reason to run 61 cards in a deck that might otherwise want 60—mana ratio. The extra card provides a good way to control the number of lands you draw without giving up much in return. You can either run an extra land or an extra spell, and either raise or lower your mana ratio slightly.

The extra card does have an effect on your ability to draw any individual card in your deck (it would be weird if it didn't). However, this effect is much smaller than the effect it has on your ability to draw lands. In fact, the effect on your ability to draw lands is three to five times larger than the effect on your ability to draw any individual card.

Let's examine the data. The first order of business is to actually look at the effect an extra card has on your ability to draw any individual card. Here is the raw data:

% Chance of Drawing a Particular Card by a Particular Draw*

60 cards

* "Draw 0" is the opening hand.

Number of Copies in the Deck
1 copy 2 copies 3 copies 4 copies 5 copies 6 copies 7 copies 8 copies
Draw 0 11.67 22.15 31.54 39.95 47.46 54.14 60.09 65.36
Draw 1 13.33 25.08 35.42 44.48 52.41 59.33 65.36 70.59
Draw 2 15.00 27.97 39.14 48.75 56.99 64.03 70.02 75.11
Draw 3 16.67 30.79 42.72 52.77 61.21 68.26 74.14 79.02
Draw 4 18.33 33.56 46.16 56.55 65.09 72.07 77.76 82.37
Draw 5 20.00 36.27 49.46 60.10 68.65 75.49 80.94 85.25
Draw 6 21.67 38.93 52.62 63.42 71.91 78.55 83.72 87.71
Draw 7 23.33 41.53 55.64 66.54 74.90 81.29 86.14 89.80
Draw 8 25.00 44.07 58.53 69.45 77.63 83.73 88.25 91.58
Draw 9 26.67 46.55 61.30 72.16 80.12 85.90 90.08 93.07
Draw 10 28.33 48.98 63.94 74.69 82.37 87.82 91.66 94.33
Draw 11 30.00 51.36 66.45 77.05 84.42 89.52 93.01 95.39
Draw 12 31.67 53.67 68.85 79.23 86.28 91.02 94.18 96.27
Draw 13 33.33 55.93 71.13 81.26 87.95 92.33 95.17 96.99
Draw 14 35.00 58.14 73.29 83.13 89.46 93.48 96.02 97.60
Draw 15 36.67 60.28 75.35 84.86 90.81 94.49 96.73 98.09
Draw 16 38.33 62.37 77.29 86.46 92.02 95.36 97.33 98.49
Draw 17 40.00 64.41 79.14 87.92 93.10 96.11 97.84 98.82
Draw 18 41.67 66.38 80.87 89.26 94.06 96.76 98.26 99.08
Draw 19 43.33 68.31 82.51 90.49 94.91 97.31 98.61 99.29
Draw 20 45.00 70.17 84.06 91.61 95.65 97.79 98.89 99.46


61 cards

Number of Copies in the Deck
1 copy 2 copies 3 copies 4 copies 5 copies 6 copies 7 copies 8 copies
Draw 0 11.48 21.80 31.08 39.40 46.84 53.49 59.41 64.67
Draw 1 13.11 24.70 34.91 43.89 51.76 58.65 64.67 69.90
Draw 2 14.75 27.54 38.59 48.12 56.31 63.33 69.33 74.45
Draw 3 16.39 30.33 42.14 52.11 60.51 67.57 73.46 78.38
Draw 4 18.03 33.06 45.54 55.87 64.39 71.38 77.10 81.77
Draw 5 19.67 35.74 48.81 59.40 67.95 74.82 80.31 84.69
Draw 6 21.31 38.36 51.94 62.71 71.22 77.90 83.12 87.19
Draw 7 22.95 40.93 54.95 65.82 74.22 80.66 85.58 89.32
Draw 8 24.59 43.44 57.82 68.73 76.96 83.13 87.73 91.14
Draw 9 26.23 45.90 60.57 71.45 79.46 85.33 89.60 92.68
Draw 10 27.87 48.31 63.20 73.99 81.75 87.29 91.22 93.98
Draw 11 29.51 50.66 65.71 76.35 83.82 89.02 92.61 95.08
Draw 12 31.15 52.95 68.10 78.55 85.70 90.55 93.82 95.99
Draw 13 32.79 55.19 70.38 80.59 87.40 91.90 94.85 96.76
Draw 14 34.43 57.38 72.55 82.49 88.94 93.09 95.73 97.39
Draw 15 36.07 59.51 74.61 84.24 90.32 94.12 96.47 97.91
Draw 16 37.70 61.58 76.56 85.86 91.56 95.03 97.11 98.34
Draw 17 39.34 63.61 78.41 87.34 92.67 95.81 97.64 98.69
Draw 18 40.98 65.57 80.16 88.71 93.66 96.49 98.09 98.97
Draw 19 42.62 67.49 81.81 89.97 94.54 97.08 98.46 99.20
Draw 20 44.26 69.34 83.37 91.11 95.32 97.58 98.77 99.38



For the numerically challenged, here is a pretty picture:

Here is its effect on your mana ratio:

% Chance of Hitting Land Drops by Number of Lands

60 cards, on the play

Drop 1 Drop 2 Drop 3 Drop 4 Drop 5 Drop 6 Drop 7 Drop 8 Drop 9 Drop 10
20 lands 95.17 82.42 63.78 44.05 27.28 15.23 7.70 3.53 1.47 0.56
21 lands 96.02 84.97 67.98 48.98 31.89 18.84 10.15 4.99 2.24 0.92
22 lands 96.73 87.24 71.90 53.85 36.71 22.86 13.04 6.84 3.30 1.46
23 lands 97.33 89.24 75.53 58.60 41.67 27.23 16.40 9.13 4.70 2.24
24 lands 97.84 90.99 78.87 63.18 46.71 31.92 20.21 11.88 6.49 3.30
25 lands 98.26 92.51 81.90 67.54 51.74 36.85 24.44 15.12 8.74 4.72
26 lands 98.61 93.82 84.63 71.64 56.70 41.95 29.05 18.85 11.47 6.56
27 lands 98.89 94.95 87.06 75.46 61.54 47.16 33.98 23.05 14.73 8.87
28 lands 99.13 95.91 89.21 78.97 66.18 52.39 39.17 27.68 18.51 11.71



60 cards, on the draw

Drop 1 Drop 2 Drop 3 Drop 4 Drop 5 Drop 6 Drop 7 Drop 8 Drop 9 Drop 10
20 lands 96.99 87.75 72.24 53.64 35.88 21.68 11.86 5.89 2.65 1.08
21 lands 97.60 89.83 76.12 58.75 41.21 26.30 15.30 8.13 3.94 1.75
22 lands 98.09 91.62 79.63 63.65 46.61 31.27 19.25 10.89 5.66 2.70
23 lands 98.49 93.15 82.79 68.28 51.99 36.52 23.68 14.19 7.86 4.03
24 lands 98.82 94.45 85.59 72.59 57.27 41.95 28.54 18.04 10.60 5.79
25 lands 99.08 95.54 88.05 76.56 62.38 47.48 33.75 22.41 13.90 8.06
26 lands 99.29 96.45 90.19 80.17 67.24 53.02 39.25 27.27 17.79 10.90
27 lands 99.46 97.20 92.03 83.41 71.81 58.46 44.92 32.55 22.25 14.34
28 lands 99.59 97.82 93.60 86.29 76.03 63.71 50.66 38.17 27.25 18.42


61 cards, on the play

Drop 1 Drop 2 Drop 3 Drop 4 Drop 5 Drop 6 Drop 7 Drop 8 Drop 9 Drop 10
20 lands 94.85 81.49 62.33 42.43 25.85 14.18 7.04 3.17 1.29 0.48
21 lands 95.73 84.09 66.53 47.29 30.30 17.60 9.30 4.49 1.98 0.80
22 lands 96.47 86.42 70.48 52.10 34.97 21.41 12.00 6.17 2.92 1.27
23 lands 97.11 88.48 74.16 56.81 39.80 25.58 15.14 8.26 4.17 1.95
24 lands 97.64 90.30 77.55 61.38 44.73 30.07 18.71 10.80 5.79 2.88
25 lands 98.09 91.89 80.65 65.75 49.68 34.83 22.70 13.79 7.82 4.14
26 lands 98.46 93.26 83.46 69.89 54.59 39.78 27.08 17.26 10.31 5.77
27 lands 98.77 94.45 85.98 73.77 59.40 44.86 31.80 21.19 13.29 7.85
28 lands 99.02 95.47 88.22 77.36 64.05 49.99 36.79 25.56 16.77 10.41


61 cards, on the draw

Drop 1 Drop 2 Drop 3 Drop 4 Drop 5 Drop 6 Drop 7 Drop 8 Drop 9 Drop 10
20 lands 96.76 86.96 70.85 51.91 34.19 20.31 10.91 5.31 2.34 0.93
21 lands 97.39 89.11 74.78 56.99 39.38 24.71 14.12 7.36 3.50 1.52
22 lands 97.91 90.97 78.36 61.88 44.66 29.48 17.83 9.89 5.04 2.36
23 lands 98.34 92.57 81.60 66.53 49.96 34.54 22.01 12.94 7.03 3.53
24 lands 98.69 93.94 84.49 70.89 55.19 39.81 26.62 16.52 9.52 5.10
25 lands 98.97 95.09 87.04 74.93 60.29 45.22 31.62 20.62 12.55 7.13
26 lands 99.20 96.06 89.28 78.63 65.17 50.66 36.91 25.20 16.13 9.69
27 lands 99.38 96.87 91.22 81.98 69.79 56.05 42.41 30.22 20.28 12.82
28 lands 99.53 97.54 92.88 84.97 74.09 61.30 48.02 35.59 24.96 16.55



And some pretty pictures again:

To show some graphs that have less action, I have selected two common land counts, 24 and 27, and graphed the 60 vs. 61 card curves for those land counts:

I will now present the data for 40 and 41 cards, in the same format. First, the draw probabilities:

40 cards

1 Copy 2 Copies 3 Copies 4 Copies 5 Copies 6 Copies
Draw 0 17.50 32.31 44.78 55.22 63.93 71.14
Draw 1 20.00 36.41 49.80 60.65 69.40 76.39
Draw 2 22.50 40.38 54.50 65.57 74.18 80.82
Draw 3 25.00 44.23 58.91 70.01 78.34 84.53
Draw 4 27.50 47.95 63.02 74.01 81.95 87.62
Draw 5 30.00 51.54 66.84 77.60 85.06 90.18
Draw 6 32.50 55.00 70.39 80.80 87.73 92.29
Draw 7 35.00 58.33 73.68 83.64 90.00 94.00
Draw 8 37.50 61.54 76.72 86.16 91.93 95.39
Draw 9 40.00 64.62 79.51 88.37 93.54 96.49
Draw 10 42.50 67.56 82.07 90.31 94.89 97.37
Draw 11 45.00 70.38 84.41 92.00 96.00 98.06
Draw 12 47.50 73.08 86.54 93.45 96.91 98.59
Draw 13 50.00 75.64 88.46 94.70 97.64 98.99
Draw 14 52.50 78.08 90.19 95.76 98.23 99.29
Draw 15 55.00 80.38 91.74 96.65 98.70 99.52



41 cards

1 Copy 2 Copies 3 Copies 4 Copies 5 Copies 6 Copies
Draw 0 17.07 31.59 43.86 54.21 62.87 70.09
Draw 1 19.59 35.73 48.97 59.76 68.50 75.53
Draw 2 22.10 39.75 53.75 64.79 73.42 80.12
Draw 3 24.61 43.64 58.23 69.33 77.71 83.96
Draw 4 27.12 47.39 62.41 73.42 81.42 87.17
Draw 5 29.64 51.02 66.29 77.09 84.62 89.83
Draw 6 32.15 54.52 69.91 80.36 87.37 92.01
Draw 7 34.66 57.89 73.25 83.27 89.71 93.78
Draw 8 37.18 61.13 76.34 85.84 91.69 95.22
Draw 9 39.69 64.24 79.18 88.11 93.35 96.37
Draw 10 42.20 67.22 81.78 90.09 94.74 97.27
Draw 11 44.72 70.07 84.16 91.81 95.88 97.99
Draw 12 47.23 72.79 86.32 93.30 96.82 98.53
Draw 13 49.74 75.38 88.27 94.58 97.57 98.95
Draw 14 52.25 77.84 90.03 95.66 98.18 99.27
Draw 15 54.77 80.18 91.60 96.58 98.66 99.50



And, here we have a pretty picture again:

And, once again, here is the data for your land-drop probabilities:

40 cards, on the play

Drop 1 Drop 2 Drop 3 Drop 4 Drop 5 Drop 6 Drop 7 Drop 8 Drop 9 Drop 10
15 lands 97.42 89.22 74.86 56.80 38.74 23.66 12.90 6.23 2.65 0.97
16 lands 98.14 91.84 80.03 64.02 46.68 30.90 18.50 9.97 4.80 2.04
17 lands 98.69 93.94 84.46 70.69 54.63 38.79 25.22 14.94 8.03 3.88
18 lands 99.09 95.59 88.17 76.68 62.31 47.06 32.89 21.20 12.54 6.77
19 lands 99.38 96.86 91.21 81.90 69.50 55.38 41.26 28.64 18.44 10.96
20 lands 99.58 97.82 93.63 86.33 75.99 63.45 50.00 37.05 25.72 16.66



40 cards, on the draw

Drop 1 Drop 2 Drop 3 Drop 4 Drop 5 Drop 6 Drop 7 Drop 8 Drop 9 Drop 10
15 lands 98.59 93.32 82.60 67.12 49.50 32.89 19.56 10.34 4.80 1.93
16 lands 99.04 95.22 86.89 73.93 57.96 41.53 27.03 15.87 8.34 3.88
17 lands 99.36 96.65 90.36 79.87 65.95 50.43 35.49 22.84 13.36 7.04
18 lands 99.58 97.71 93.10 84.89 73.21 59.20 44.58 31.09 19.96 11.71
19 lands 99.74 98.48 95.19 88.99 79.58 67.48 53.88 40.29 28.05 18.08
20 lands 99.84 99.02 96.76 92.24 84.96 74.97 62.95 50.00 37.38 26.16



41 cards, on the play

Drop 1 Drop 2 Drop 3 Drop 4 Drop 5 Drop 6 Drop 7 Drop 8 Drop 9 Drop 10
15 lands 97.07 88.04 72.70 54.02 35.93 21.34 11.28 5.28 2.16 0.76
16 lands 97.86 90.82 78.01 61.20 43.58 28.08 16.31 8.51 3.96 1.62
17 lands 98.46 93.07 82.62 67.93 51.34 35.52 22.43 12.88 6.69 3.12
18 lands 98.91 94.87 86.54 74.05 58.94 43.43 29.52 18.45 10.56 5.50
19 lands 99.24 96.27 89.80 79.48 66.17 51.52 37.38 25.19 15.71 9.02
20 lands 99.48 97.35 92.45 84.17 72.82 59.51 45.74 32.95 22.17 13.88



41 cards, on the draw

Drop 1 Drop 2 Drop 3 Drop 4 Drop 5 Drop 6 Drop 7 Drop 8 Drop 9 Drop 10
15 lands 98.36 92.42 80.71 64.35 46.35 29.97 17.29 8.84 3.96 1.53
16 lands 98.87 94.48 85.22 71.27 54.66 38.16 24.12 13.71 6.96 3.12
17 lands 99.23 96.06 88.92 77.41 62.63 46.74 31.98 19.95 11.28 5.73
18 lands 99.49 97.25 91.89 82.68 70.02 55.35 40.59 27.47 17.06 9.66
19 lands 99.67 98.12 94.22 87.09 76.63 63.64 49.57 36.03 24.30 15.13
20 lands 99.79 98.75 96.00 90.66 82.33 71.32 58.53 45.26 32.83 22.22



And here are some more graphs:

Once again, I've selected common land counts—16, 17, 18—and graphed them independently.


Conclusion

The key here is to look at the spacing between the red and blue curves. It is evident that the extra card has a minimal effect on draw probabilities, especially past the first three or four draws. In fact, its effect is nearly always less than 1%. However, it is clear that by using the extra card, a player can gain extra resolution in their land-drop frequencies, allowing more fine-tuning for mana ratios. The effect of the extra card is in the 2% to 3% range for 60 cards, and the 3% to 5% range for 40 cards. If you don't use the potential of the extra card, you are left fine-tuning your mana ratios in larger increments. In fact, your ability to fine-tune is about half as good.

Now, let me ask you—do you lose more games to not drawing a specific card, or to drawing lands at the wrong time?

Chingsung Chang
Conelead most everywhere and on MTGO
Khan32k5 at gmail dot com