Modeling the Expected Value of a Sealed Card Box (and Where the Number Quietly Lies)
A friend messaged me a photo of a sealed booster box last month with one question: "worth it?" He'd already decided, really. The chase card in that set was all over his feed, so the box felt like a good deal. I asked him to send me the pull rates instead of the hype, and we spent twenty minutes turning "worth it?" into something we could actually compute. That exercise is a small, self-contained data problem. It's also a good example of how a clean-looking model can hand you a confident number that doesn't survive contact with reality. If you like building little estimators, this one is worth doing carefully, because the interesting part isn't the formula. It's everything the formula assumes. The formula is the easy part Expected value of a box is a weighted sum. Each card you can pull has a probability and a market value, and you multiply the two across every slot the box gives you. That's it. Undergrad probability. Here's a stripped-down version for a hypothetical set. I'm using made-up numbers so nobody mistakes this for real pull data — the point is the shape of the computation, not the specific set. # One "hit slot" in a box: probabilities cover the full outcome space. # Values are illustrative market estimates in USD. hit_table = [ { " name " : " Alt-art chase " , " p " : 0.0125 , " value " : 180.00 }, { " name " : " Secret rare " , " p " : 0.030 , " value " : 45.00 }, { " name " : " Full-art rare " , " p " : 0.100 , " value " : 8.00 }, { " name " : " Standard hit " , " p " : 0.400 , " value " : 0.55 }, { " name " : " No notable hit " , " p " : 0.4575 , " value " : 0.06 }, ] assert abs ( sum ( row [ " p " ] for row in hit_table ) - 1.0 ) < 1e-9 ev_per_slot = sum ( row [ " p " ] * row [ " value " ] for row in hit_table ) hit_slots_per_box = 36 # e.g. one meaningful slot per pack ev_box = ev_per_slot * hit_slots_per_box print ( f " EV per slot: $ { ev_per_slot : . 2 f } " ) # $4.65 print ( f " EV per box: $ { ev_box : . 2 f } " ) # $167.31 The box costs $150 sea