Below is one way to break down the calculation. In Lorcana you roll with a 60‐card deck that contains 4 copies of “Belle” and 4 copies of “Pawpsicle.” At the start you draw 7 cards. Then—using your one mulligan opportunity—you may keep any “key” (i.e. Belle or Pawpsicle) cards you already have and put all the rest on the bottom of your deck, drawing new cards until you again have 7 cards in hand. (In other words, if you’re missing one or both of your key cards, you discard every “non‐key” card and draw that many replacements from the remainder of the deck.) Under the assumption that you follow this optimal strategy, we can split the outcomes into four cases.
Step 1. Find the chance you see each key in the initial 7 without mulligan.
Let’s define the events:
• A = “at least one Belle is in your 7‐card hand”
• B = “at least one Pawpsicle is in your 7‐card hand”Because there are 4 copies of a given key in a 60‑card deck, the chance to see no copy of (say) Belle in a 7‑card draw is