Booster Box Expected Value Has Error Bars

Booster box expected value is a single number resting on three inputs, none of which is measured to the precision the number implies. Across the 27 English sets where we track both an unambiguous 36-pack booster box and pull rates traceable to a published study, the cards in a box are worth a median 0.380 of the box's own ask. Traceable is the operative word and it is weaker than it sounds: only 7 of those 27 sets carry any rate measured on that set, and only 3 are measured all the way through. The other 20 inherit an era baseline measured on a different set, which Pokemon Pull Rates by Set takes apart set by set. That figure is not the finding. The finding is how much it moves when you change one assumption at a time: the published odds' own 95% confidence intervals shift it by a median of 3.0% across the panel, or 5.2% across the fifteen sets that actually have an interval to shift, while the single decision to count or discard cards worth under $5 shifts it by 47.5%.

An expected value per box is the product of three things we do not know exactly. The pull rate for each tier, which comes from a published sample and carries a confidence interval when the publisher bothered to print one. The price of the cards that fall out, which on this site is a TCGplayer market ask rather than a sold price. And what a seller can actually turn those cards into, which is neither of the first two, because 360 cards from one box include several hundred that no buyer will purchase individually at any price.

We compute the expectation in closed form rather than by simulation, over exactly the slot plans, tier odds and card pools the site's pack simulator uses. The point of the closed form is that it is exactly decomposable: every dollar of expected value is attributable to one tier, so moving one input's assumption shows its contribution directly. To check the closed form is the same model, we ran the site's Monte Carlo engine against it on five sets.

The largest gap in that run is 1.57%, which is Monte Carlo noise at 3,000 boxes rather than a disagreement about the model; re-running the simulator gives gaps of a similar size in either direction, so read 1.57% as a sample of the noise rather than a bound on it. Everything below is the closed form.

The two coverage columns differ only in assumption. The ask column values every card that falls out of the box at its TCGplayer market ask, which is the convention every published box-EV number uses. The second column keeps only cards worth $5 or more and multiplies by a stated 0.85 realisation factor, which stands in for marketplace fees, shipping and the discount a seller accepts to move a card promptly. Neither column is the truth. The distance between them is the honest width of the answer.

No set reaches 1.0 on either basis, which is the result 260,000 Simulated Boxes already established by simulation on a 26-set panel. This study is not re-deriving that. It is measuring how far the answer can be pushed by the assumptions underneath it, which is the part a reader needs before quoting any of these numbers.

The odds are the input everyone argues about and the one that matters least. Even at their published extremes they move the answer by 3.0% on a typical set in the panel and never past 10.5%. That 3.0% needs one caveat, because twelve of the 27 sets contribute an exact zero to it, and a zero there means no interval was published rather than a narrow one. Among the fifteen sets that do carry an interval the median swing is 5.2% and the range is 2.9% to 10.5%. Either way the odds remain the smallest of the four axes, by a wide margin. What moves the answer is bookkeeping. Nearly half the ask-basis expected value of a box is in cards worth less than five dollars, which exist, are genuinely worth what the ask says in a bulk lot, and are not money in any practical sense for a single box opened by a single person.

Read that table one column at a time. Each column is an independent median across the 27 sets, so the two columns describe different sets and no row is a decomposition of any single box: the shares sum to 96.0% rather than 100%, and the dollars sum to $124.32 against a median box expectation of $133.13. The two rows below $5 also sum to 49.9%, while the sensitivity table's median of the per-set under-$5 share is 47.5%; both are honest, they are a sum of medians and a median of sums, and neither is the other. Two fifths of a box's paper value is in cards worth under a dollar. That is the single most useful number in this study, because it explains why box EV calculations published elsewhere look generous and why the people who run them are disappointed. It is also why the floor under bulk matters, which Bulk's Hidden Floor measured directly.

The Sword & Shield era is most of the problem. The Lost Origin study that our SWSH baseline rests on publishes point estimates without intervals, and that accounts for 11 of the 12 boxes here on which we cannot state how wrong the odds might be. The twelfth is Scarlet & Violet, whose rates come from a secondary report of that set's own study and which publishes point estimates too, so it is a Scarlet & Violet gap rather than a Sword & Shield one. Of the 12 SWSH boxes in the panel, Evolving Skies is the one that does carry intervals, and the seven SWSH tier rates with an interval all come from that separate study. An absent interval is not a narrow one, and we print "none published" rather than assume the SV era's widths apply.

The card prices are ASKS. Every coverage ratio here would fall if sold prices were used, and by an amount this study does not measure. The 0.85 realisation factor is a stated convention, not a measurement, chosen to sit near the 15% cushion this site already demands before trusting a raw ask, measured in The Ask Is Lying to You. Treat the second coverage column as an illustration of sensitivity rather than as a forecast of proceeds.

The engine values an unpriced card at $0, which is a downward bias. On these 27 sets the unpriced share of pooled cards is under 1% everywhere, so the bias is small here, but it would not be on an older or thinner set. Tiers whose rates are not published are never rolled, which means a set with an unpublished god-pack or gallery rate has a small, unquantified slice of its contents missing from the expectation entirely.

The panel is 27 sets because it requires three things at once: a pack model entry, resolvable published odds, and a single unambiguous 36-pack booster box SKU. Resolvable published odds does not mean the odds were measured on that set: 20 of the 27 inherit an era baseline measured elsewhere, 7 carry at least one rate of their own, and only 3 are measured throughout. That is a selection worth stating plainly, because a reader can reasonably take published pull rates to mean somebody opened packs of this set. Eight sets in our pack model have no such box: 151, Paldean Fates, Shrouded Fable, Prismatic Evolutions, Black Bolt, White Flare, Crown Zenith and Ascended Heroes. They are absent here because our catalog holds no single unambiguous 36-pack box SKU under those set names, which for most of them is because the set was not sold that way. Set composition counts used to build the pools are catalog facts rather than findings of ours.

Finally, the box ask is one price on one day from one marketplace. Sealed asks are refreshed far less often than most readers assume, which is measured directly in Sealed vs Singles. On an expensive vintage box the ask can sit unchanged for the whole of a 23-day window.

A matched panel of realised sale proceeds from opened boxes, covering the same sets on the same dates, that landed above the ask-basis contents value would overturn the entire direction of this study, since every assumption we test only pushes the number down. More narrowly, a published pull-rate study showing that draws inside a rare tier are materially non-uniform would invalidate the mean-of-pool convention and with it the 0.380 median, in a direction set by which cards are short-printed. And if a future study published confidence intervals for the Sword & Shield era that turned out to be wider than 10.5% of expected value, the claim that odds are the least important input would need revisiting for that era.

Universe: the 35 sets in SIMULATABLE_SETS in src/server/pack-sim.ts that have resolvable published tier odds in src/content/pack-model.json and non-empty catalog pools, intersected with the sets for which sealed_products holds a Booster Box SKU with a pack count of 36 whose name is exactly "<set> Booster Box". That intersection is 27 sets. The exact-name rule matters and is not a formality: Journey Together and Mega Evolution each hold TWO non-lot 36-pack Booster Box SKUs, a standard and an Enhanced, and the rule picks the standard box in both. A rule phrased as "exactly one such SKU" would return 25 sets, not 27. Expected value is computed in closed form over buildSlotPlans() and resolveTierOdds() from src/server/pack-sim.ts, unchanged: for each slot, each rolled tier contributes its published per-pack probability times the mean market price of its catalog pool, and the residual probability falls to that slot's default tier through the same defaultSplit the simulator uses. Structural slots (commons, uncommons) contribute their count times the pool mean. Per box is per pack times 36. Card prices are marketPriceByCardId(), the same resolver every card page uses; an unpriced card contributes $0, matching the site's Monte Carlo engine, and the unpriced share is under 1% on every set here. DEDUPLICATION: 1,212 TCGplayer product ids in our catalog are claimed by two card rows each, so any sum over cards that ignores the join key overstates itself. We checked every one of the 35 pools: no pool carries a duplicate card id, and no pool shares a tcg_id internally, because every one of those 1,212 collisions pairs a mainline row with a trainer-kit or foreign-language twin in a different set. Zero rows collapsed here, and the check is in apps/web/scripts/sealed-b5-dedupe.mts. Box prices are read directly from sealed_products on 2026-09-22 rather than through the cached boxprice:v4 resolver, which was found serving one stale figure. Tier counts in the confidence-interval table are ROLLED tiers: those that appear as an outcome in a slot plan AND have at least one card in that set's catalog pool. That excludes 11 Scarlet & Violet ACE SPEC and Hyper Rare rates for sets containing no such cards, and 9 Sword & Shield Radiant rates for sets predating Radiant Pokemon. Study 5 counts resolved rates instead, which is a larger number, and says so. The odds swing moves each tier's published rate by its own 95% half-width, valued as 36 times that half-width times the absolute price gap between that tier's pool mean and its slot's default tier pool mean, combined across tiers in quadrature; tiers whose published rate carries no interval contribute nothing and are reported as "none published" rather than zero. The $5 threshold and the 0.85 realisation factor are stated assumptions, not measurements. Medians take the upper-middle value on even samples. Scripts: apps/web/scripts/sealed-b-boxev.mts, sealed-b2-stack.mts, sealed-b3-summary.mts and sealed-b4-reconcile.mts, with every figure and query in apps/web/.scratch-sealed/figures.json.