Mulligan

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During setup, each player, starting with the first player, has one opportunity to mulligan their starting hand. This is done by shuffling the starting hand back into the deck and drawing a new starting hand with one fewer card in it.

After a player chooses to mulligan, that player must keep the new starting hand.

If a player is using a deck that has chains applied to it at the start of the game and takes a mulligan, they do not shed a chain from the mulligan, but do draw one fewer card than they had before the mulligan as per the normal mulligan rules.

Commentary

If you are the first player, drawing 7 cards gives you the following probabilities for your hand shape:

House 1 House 2 House 3 Probability
3 2 2 34.44
4 2 1 28.18
3 3 1 20.87
4 3 0 7.83
5 1 1 4.10
5 2 0 3.76
6 1 0 .80
7 0 0 .03

After the mulligan (or if you're the second player) you have the following probabilities of hand shape:

House 1 House 2 House 3 Probability
3 2 1 53.67
2 2 2 14.76
4 1 1 10.98
4 2 0 10.06
3 3 0 7.45
5 1 0 2.93
6 0 0 .14

And if you're the second player taking a mulligan, you have the following:

House 1 House 2 House 3 Probability
2 2 1 41.60
3 1 1 25.21
3 2 0 23.11
4 1 0 9.45
5 0 0 .63

The question often comes up "How likely am I to redraw cards that I just put back into my deck after using a mulligan?". In the table below, you'll see the results of 10,000,000 6 card draws followed by 5 card draws of three deck makeups[1].

# redrawn All Unique 2 sets of 2 cards 1 set of 3 cards
0 37.81294 34.25509 33.09266
1 43.61144 43.80313 43.60717
2 16.14783 18.5513 19.40653
3 2.30691 3.1727 3.60981
4 0.1194 0.21365 0.27703
5 0.00148 0.00413 0.0068%
  1. Note that these numbers are derived using the Fisher-Yates shuffling algorithm, which is the same algortihm used on The Crucible Online.
    In the "All Unique" model, every card in the deck is unique; there is no more than one copy of each card in the deck. In the "2 sets of 2 cards" model, there are duplicates of two cards in the deck (e.g. 2 copies of Bad Penny and 2 copies of Yxilo Bolter), and 1 copy of all other cards in the deck. In the "1 set of 3 cards" model, one card has 3 copies in the deck (e.g. 3 copies of Bad Penny) while all other cards in the deck have just 1 copy. Numbers courtesy of SkyJedi