Template:MulliganCommentary
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 table below represents the probability of drawing at least one copy of a card in your initial draw based on the number of copies of the specific card in your deck. For example, if I have 3 copies of Urchin in my deck, I will draw at least one copy of Urchin 48.82% of the time with a 7 card starting hand. If I mulligan hoping to get an Urchin-free hand, a copy of Urchin will still show up 43.14% of the time. Similarly, going second and hoping to use Urchin to steal an Æmber gained by the first player, if Urchin didn't show up in my initial 6 card hand, there is a 37.04% chance it will show up after the mulligan.
| Copies of a Card in your deck | ||||
|---|---|---|---|---|
| Starting Hand Size | 1 | 2 | 3 | 4 |
| 7 | 19.44 | 35.56 | 48.82 | 59.68 |
| 6 | 16.67 | 30.95 | 43.14 | 53.48 |
| 5 | 13.89 | 26.19 | 37.04 | 46.58% |
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. 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.
| # 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% |