Avg # of Arcana |
Min |
Max |
Mean |
Wt. Mean |
Dice Mean |
vs Dice |
||||

Odds for a Two Card Hand |
0.571428571428571 |
7.857142857142850 |
11 |
-3.14285714285715 |
||||||

Minor Arcana | 0 | 50.64935064935060% | 0.000000000000000 | 2 | 20 | 11 | 5.571428571428570 | |||

Minor Arcana | 1 | 41.55844155844150% | 0.415584415584415 | 1 | 10 | 5.5 | 2.285714285714280 | |||

Minor Arcana | 2 | 7.79220779220779% | 0.155844155844156 | 0 | 0 | 0 | 0.000000000000000 | |||

Odds for a Three Card Hand |
0.857142857142857 |
11.785714285714300 |
16.5 |
-4.71428571428573 |
||||||

Minor Arcana | 0 | 35.64213564213560% | 0.000000000000000 | 3 | 30 | 16.5 | 5.880952380952370 | |||

Minor Arcana | 1 | 45.02164502164500% | 0.450216450216450 | 2 | 20 | 11 | 4.952380952380950 | |||

Minor Arcana | 2 | 17.31601731601730% | 0.346320346320346 | 1 | 10 | 5.5 | 0.952380952380951 | |||

Minor Arcana | 3 | 2.02020202020202% | 0.060606060606061 | 0 | 0 | 0 | 0.000000000000000 | |||

Odds for a Four Card Hand |
1.142857142857140 |
15.714285714285700 |
22 |
-6.28571428571429 |
||||||

Minor Arcana | 0 | 24.88224563696260% | 0.000000000000000 | 4 | 40 | 22 | 5.474094040131770 | |||

Minor Arcana | 1 | 43.03956002069210% | 0.430395600206921 | 3 | 30 | 16.5 | 7.101527403414200 | |||

Minor Arcana | 2 | 25.48395001225190% | 0.509679000245038 | 2 | 20 | 11 | 2.803234501347710 | |||

Minor Arcana | 3 | 6.09872307985516% | 0.182961692395655 | 1 | 10 | 5.5 | 0.335429769392034 | |||

Minor Arcana | 4 | 0.49552125023823% | 0.019820850009529 | 0 | 0 | 0 | 0.000000000000000 | |||

Odds for a Five Card Hand |
1.428571428571430 |
19.642857142857100 |
27.5 |
-7.85714285714287 |
||||||

Minor Arcana | 0 | 17.22617005635870% | 0.000000000000000 | 6 | 49 | 27.5 | 4.737196765498640 | |||

Minor Arcana | 1 | 38.28037790301940% | 0.382803779030194 | 4 | 40 | 22 | 8.421683138664270 | |||

Minor Arcana | 2 | 31.03814424569140% | 0.620762884913828 | 3 | 30 | 16.5 | 5.121293800539080 | |||

Minor Arcana | 3 | 11.43510577472840% | 0.343053173241852 | 2 | 20 | 11 | 1.257861635220120 | |||

Minor Arcana | 4 | 1.90585096245474% | 0.076234038498190 | 1 | 10 | 5.5 | 0.104821802935011 | |||

Minor Arcana | 5 | 0.11435105774728% | 0.005717552887364 | 0 | 0 | 0 | 0.000000000000000 | |||

Odds for a Six Card Hand |
1.714285714285710 |
23.571428571428500 |
33 |
-9.42857142857148 |
||||||

Minor Arcana | 0 | 11.82188141122650% | 0.000000000000000 | 8 | 58 | 33 | 3.901220865704740 | |||

Minor Arcana | 1 | 32.42573187079290% | 0.324257318707929 | 6 | 49 | 27.5 | 8.917076264468050 | |||

Minor Arcana | 2 | 33.77680403207590% | 0.675536080641518 | 4 | 40 | 22 | 7.430896887056700 | |||

Minor Arcana | 3 | 17.04054978194820% | 0.511216493458446 | 3 | 30 | 16.5 | 2.811690714021450 | |||

Minor Arcana | 4 | 4.37224632563145% | 0.174889853025258 | 2 | 20 | 11 | 0.480947095819459 | |||

Minor Arcana | 5 | 0.53812262469310% | 0.026906131234655 | 1 | 10 | 5.5 | 0.029596744358121 | |||

Minor Arcana | 6 | 0.02466395363177% | 0.001479837217906 | 0 | 0 | 0 | 0.000000000000000 | |||

Odds for a Seven Card Hand |
2.000000000000000 |
27.500000000000000 |
38.5 |
-11.00000000000000 |
||||||

Minor Arcana | 0 | 8.03887935963407% | 0.000000000000000 | 10 | 67 | 38.5 | 3.094968553459120 | |||

Minor Arcana | 1 | 26.48101436114750% | 0.264810143611475 | 8 | 58 | 33 | 8.738734739178670 | |||

Minor Arcana | 2 | 34.04701846433250% | 0.680940369286650 | 6 | 49 | 27.5 | 9.362930077691440 | |||

Minor Arcana | 3 | 22.06751196762290% | 0.662025359028687 | 4 | 40 | 22 | 4.854852632877040 | |||

Minor Arcana | 4 | 7.75345015078644% | 0.310138006031458 | 3 | 30 | 16.5 | 1.279319274879760 | |||

Minor Arcana | 5 | 1.46907476541217% | 0.073453738270609 | 2 | 20 | 11 | 0.161598224195339 | |||

Minor Arcana | 6 | 0.13811814033790% | 0.008287088420274 | 1 | 10 | 5.5 | 0.007596497718584 | |||

Minor Arcana | 7 | 0.00493279072635% | 0.000345295350845 | 0 | 0 | 0 | 0.000000000000000 | |||

Odds for an Eight Card Hand |
2.285714285714280 |
31.428571428571400 |
44 |
-12.57142857142860 |
||||||

Minor Arcana | 0 | 5.41393916056989% | 0.000000000000000 | 12 | 76 | 44 | 2.382133230650750 | |||

Minor Arcana | 1 | 20.99952159251350% | 0.209995215925135 | 10 | 67 | 38.5 | 8.084815813117700 | |||

Minor Arcana | 2 | 32.42573187079290% | 0.648514637415858 | 8 | 58 | 33 | 10.700491517361700 | |||

Minor Arcana | 3 | 25.94058549663430% | 0.778217564899029 | 6 | 49 | 27.5 | 7.133661011574430 | |||

Minor Arcana | 4 | 11.70929206445290% | 0.468371682578116 | 4 | 40 | 22 | 2.576044254179640 | |||

Minor Arcana | 5 | 3.03808658969591% | 0.151904329484796 | 3 | 30 | 16.5 | 0.501284287299825 | |||

Minor Arcana | 6 | 0.43972305903493% | 0.026383383542096 | 2 | 20 | 11 | 0.048369536493843 | |||

Minor Arcana | 7 | 0.03221414351904% | 0.002254990046333 | 1 | 10 | 5.5 | 0.001771777893547 | |||

Minor Arcana | 8 | 0.00090602278647% | 0.000072481822918 | 0 | 0 | 0 | 0.000000000000000 | |||

Odds for a Nine Card Hand |
2.571428571428560 |
35.357142857142800 |
49.5 |
-14.14285714285720 |
||||||

Minor Arcana | 0 | 3.60929277371325% | 0.000000000000000 | 15 | 84 | 49.5 | 1.786599922988060 | |||

Minor Arcana | 1 | 16.24181748170960% | 0.162418174817096 | 12 | 76 | 44 | 7.146399691952220 | |||

Minor Arcana | 2 | 29.53057723947210% | 0.590611544789442 | 10 | 67 | 38.5 | 11.369272237196800 | |||

Minor Arcana | 3 | 28.37251538694370% | 0.851175461608311 | 8 | 58 | 33 | 9.362930077691420 | |||

Minor Arcana | 4 | 15.80754428701150% | 0.632301771480460 | 6 | 49 | 27.5 | 4.347074678928160 | |||

Minor Arcana | 5 | 5.26918142900384% | 0.263459071450192 | 4 | 40 | 22 | 1.159219914380840 | |||

Minor Arcana | 6 | 1.04434226520797% | 0.062660535912478 | 3 | 30 | 16.5 | 0.172316473759315 | |||

Minor Arcana | 7 | 0.11778296224150% | 0.008244807356905 | 2 | 20 | 11 | 0.012956125846565 | |||

Minor Arcana | 8 | 0.00679517089855% | 0.000543613671884 | 1 | 10 | 5.5 | 0.000373734399420 | |||

Minor Arcana | 9 | 0.00015100379775% | 0.000013590341797 | 0 | 0 | 0 | 0.000000000000000 | |||

Odds for a Ten Card Hand |
2.857142857142850 |
39.285714285714200 |
55 |
-15.71428571428580 |
||||||

Minor Arcana | 0 | 2.38059736138534% | 0.000000000000000 | 18 | 92 | 55 | 1.309328548761940 | |||

Minor Arcana | 1 | 12.28695412327910% | 0.122869541232791 | 15 | 84 | 49.5 | 6.082042291023150 | |||

Minor Arcana | 2 | 25.91779385379200% | 0.518355877075840 | 12 | 76 | 44 | 11.403829295668500 | |||

Minor Arcana | 3 | 29.32114052146160% | 0.879634215643848 | 10 | 67 | 38.5 | 11.288639100762700 | |||

Minor Arcana | 4 | 19.61929255480150% | 0.784771702192060 | 8 | 58 | 33 | 6.474366543084500 | |||

Minor Arcana | 5 | 8.07193750826121% | 0.403596875413060 | 6 | 49 | 27.5 | 2.219782814771830 | |||

Minor Arcana | 6 | 2.05535445812206% | 0.123321267487324 | 4 | 40 | 22 | 0.452177980786853 | |||

Minor Arcana | 7 | 0.31742925994163% | 0.022220048195914 | 3 | 30 | 16.5 | 0.052375827890369 | |||

Minor Arcana | 8 | 0.02819273032376% | 0.002255418425901 | 2 | 20 | 11 | 0.003101200335614 | |||

Minor Arcana | 9 | 0.00128513870422% | 0.000115662483380 | 1 | 10 | 5.5 | 0.000070682628732 | |||

Minor Arcana | 10 | 0.00002248992732% | 0.000002248992732 | 0 | 0 | 0 | 0.000000000000000 |

*[Full disclosure: the weighted means are not absolutely accurate; as card draws are dependant events and dice rolls are independent events, the distribution curve (and thus the "true" average) is going to be slightly different. However, as I expect any changes to be in or around the 7th or 8th decimal places, the overall arguments herein are not changed.]*

As you can see, the use of cards imposes a penalty of between 1.5714285714285 and 1.5714285714286 per die. A change in result of around +/-5 will result in a change in the number of Raises the roll produces. Thus at pools of four, seven, and ten, a player using the card option is likely to lose a Raise against a player rolling the same number of dice. Fortunately, the solution to this issue is obvious. At pools of four and seven, the average number of Minor Arcana per hand increases by one at the same time the average result of a hand produces one fewer Raise.

*Proposed Rule Change: Any Minor Arcana which is not played or retained (via the Ace in the Hole rule) may be matched with any numerical card to create a Raise. *

*Alternate Proposed Rule Change: The above rule is treated as a Rank Bonus at Rank either Rank 2 or Rank 3. *

The logic behind the alternate is that, for pools of two, this rule change creates an slightly outsized likelihood of a Raise (69.41558441558433% with cards as opposed to 64% with dice).

*[Full disclosure Part II: I hate statistics, which is why I am struggling with the distribution curves for the card hands. If anyone knows off hand how to calculate the probability of a given pip total, please shoot me a ping.]*

**“Every normal man must be tempted at times to spit on his hands, hoist the black flag, and begin to slit throats.” ****- H.L. Mencken**

This looks great!

I have a few questions and requests (if I may :P ).

Which software are you using to calculate this?

Could you add the weighted mean deviations for cards and dice?

Would it be possible for you to calculate the same stuff for when two and three players are drawing from the same deck? Do you think that the average result would increase, decrease or remain the same with a higher number of players?

I really like your proposed rule. Would the Raise obtained with "numerical card + Arcana" count double when is possible to obtain two Raises with a 15?

How much the mean would increase if you could get Raises by combining two Minor Arcana cards?

There are special effects that add or may add Raises to the total in particular situations (Dame of Cups, Queen of Coins and King of Coins). Given that the cards can be used when drawn, how much would those effects affect the mean values?

I know it is hard to say, as it depends a lot on the players, but it would be interesting to see how the mean changes when the players keep high value cards (lets say 8-10) when they hand is higher than the mean and use them when they hand is lower.

Thanks for the analysis, Doctor.

Which software are you using to calculate this?These calculations were done the hard way, with a TI-82 and combinatronics formulas.

Could you add the weighted mean deviations for cards and dice?I can add the deviations for dice, but I am still figuring out the distribution curves and deviations for the cards.

# DiceDeviation1

2.87

2

4.06

3

4.97

4

5.74

5

6.42

6

7.04

7

7.6

8

8.12

9

8.62

10

9.08

Would it be possible for you to calculate the same stuff for when two and three players are drawing from the same deck? Do you think that the average result would increase, decrease or remain the same with a higher number of players?While these calculations are possible, I don't think they would be useful. With multiple players drawing, for example, each of the 63 possible outcomes for the first player would generate 63 possible outcomes for the second, meaning that there would be 3969 probabilities to calculate for player 2. The average result for player 2 would depend entirely on the draw for player 1.

I really like your proposed rule. Would the Raise obtained with "numerical card + Arcana" count double when is possible to obtain two Raises with a 15?I don't think it would, only because the odds of getting a total of 15 or above on two cards are significantly lower than getting a 10 or above. I would consider allowing it with any

twonumerical cards, however, as the odds of getting 15 or above on three cards are fairly good.How much the mean would increase if you could get Raises by combining two Minor Arcana cards?I'll crunch these numbers shortly. The issue is that the number of Raises from a given hand (or roll) is not entirely dependent on the total result of the roll (e.g. a roll of 9,9,9,9 totals 36, but only yields two Raises, while a roll of 10,10,8,8 has the same total but yields three Raises). There is not, so far, a good way to model Raises. On the whole, however, you average a little under one Raise per two cards (based on some rudimentary calculations) so I expect that the mean number of Raises would increase measurably if you allowed two Arcana to be combined for a Raise.

There are special effects that add or may add Raises to the total in particular situations (Dame of Cups, Queen of Coins and King of Coins). Given that the cards can be used when drawn, how much would those effects affect the mean values?Only two of the 16 Minor Arcana directly grant Raises, the Dame of Cups and the Queen of Swords, and while the odds of getting either change based on the number of face cards you draw, you have a 12.5% chance to draw one of the two on your first face card. The problem here is "given that the cards can be used when drawn:" the Queen of Swords (among others) specifies that it must be played before a Risk, meaning it cannot be played for the Risk it is drawn on. This leaves the Dame of Cups, which can only be played " after you make Raises for a romance-themed Risk;" calculating how often that happens would be needed to give a meaningful statement about how much the card affects mean values.

I know it is hard to say, as it depends a lot on the players, but it would be interesting to see how the mean changes when the players keep high value cards (let's say 8-10) when they hand is higher than the mean and use them when the hand is lower.I think Hero Point economy would limit that behavior substantially. A more interesting question, I think, would be how often players keep Minor Arcana.

“Every normal man must be tempted at times to spit on his hands, hoist the black flag, and begin to slit throats.”- H.L. Mencken