The Dice Challenge
When Personal Challenge and Turncoat Dragon Cards are played, the two players are involved in a sort of "dice challenge" : they are both required to roll a possibly different number of dice and count the outcomes of one particular symbol, Honor&Fortune for Turncoat and Swords for Personal Challenge. The player who rolls more symbols wins the challenge, gaining the benefits illustrated in the card. I observed that, especially with Personal Challenge, different players tend to manage it differently. Some seem to play it at the first occasion, other discard it for extra H&F at the end of the turn. The effect of these cards can be important, and can contribute significantly to the success of an attack or the victory of a scenario. Thus, I start to wonder about the actual probability to win such a challenge. Whence the table below. Each row is labeled with the number of dice of the "stronger" player, that is the player rolling more (or equal) dice, while columns are labeled with the number of dice of the weakest player. Two figures are reported: the first is the probability to win, the second the probability of a tie.
Since the minimum allowed number of Command Cards is three, and since I've never seen a scenario in which a player is assigned more than six cards, the table below should cover all the cases actually met in game play.
The pattern that emerges is interesting. Also when the difference of dice is large, the probability of victory of the strongest player is never overwhelming, reaching a maximum of 68% (10 dice vs. 3 dice). The reason is that the probability to tie the rolls is always sizable, more than 20%. When the number of dice is the same, the probability of victory of one side is around 1/3 with 5 or 6 dice, it slightly increases with more dice but it's only 25% with 3 dice per side.
Does the +2 dice bonus of playing the Personal Challenge card constitute a strong advantage? To some extent. With 7 dice vs 5 dice, the stronger player has the 44% probability to win, against the 25% of the weakest. This difference might seem large, but it means that in 1 out of 4 times the card is played, the player who played it does actually lose the challenge.
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3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
| 3 |
0.27/0.46 |
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| 4 |
0.35/0.42 |
0.30/0.40 |
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| 5 |
0.42/0.38 |
0.37/0.37 |
0.33/0.35 |
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| 6 |
0.48/0.35 |
0.43/0.34 |
0.38/0.33 |
0.34/0.32 |
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| 7 |
0.54/0.31 |
0.49/0.31 |
0.44/0.31 |
0.39/0.30 |
0.35/0.29 |
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| 8 |
0.59/0.28 |
0.54/0.29 |
0.49/0.29 |
0.44/0.29 |
0.40/0.28 |
0.36/0.27 |
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| 9 |
0.64/0.25 |
0.59/0.26 |
0.54/0.27 |
0.49/0.27 |
0.45/0.27 |
0.41/0.26 |
0.37/0.26 |
|
| 10 |
0.68/0.23 |
0.63/0.24 |
0.58/0.25 |
0.54/0.25 |
0.49/0.25 |
0.45/0.25 |
0.41/0.25 |
0.38/0.24 |