Probability Management: Shop Odds, Copies Seen, and Expected Roll Value

At the advanced level, "should I roll?" is an expected-value question with real inputs. You won't do the arithmetic exactly at the table, but you should have internalised the shape of it well enough to know when a roll is +EV and when you're lighting gold on fire.

The per-slot hit chance

Each shop slot rolls a rarity from the level table, then a random unit of that rarity from the copies still in the pool. Your chance that a given slot is your specific unit is:

P(slot) ≈ (rarity odds at your level) × (copies of your unit left in pool) ÷ (total copies of that rarity left in pool)

A shop shows six slots, so P(at least one this roll) ≈ 1 − (1 − P(slot))⁶. The two levers you control are your level (rarity odds) and, indirectly, the pool state (by scouting and by timing).

Worked example

You want a specific Rare at level 7 (Rare odds ≈ 35% per slot). There are 18 copies of each Rare; suppose ~13 different Rares are still meaningfully in the pool (~230 total copies), and you've already bought 3 of your target, so 15 remain. P(slot) ≈ 0.35 × (15 ÷ 230) ≈ 2.3%. Per six-slot roll ≈ 1 − 0.977⁶ ≈ 13%. So on average ~7–8 rolls (≈8 gold) per copy. Need three more copies for a 3★? Budget ~24 gold and expect variance either way.

Now suppose two other players are also on Rares and the tier is drained to ~120 copies with your target down to 8 left: P(slot) ≈ 0.35 × (8 ÷ 120) ≈ 2.3%… similar — unless your specific unit is the contested one, in which case its share collapses and the roll is much worse. This is why counting outs matters more than knowing the base odds.

Tracking copies seen

Every copy of your unit you see in a shop and don't buy is information: it means the pool still has copies flowing. A long dry streak with none seen means either the RNG is spiking against you or the copies are on someone's board/bench. Cross-reference with a scout. Roughly:

• Seeing your unit every couple of rolls but not enough to combine → keep rolling, it's coming.
• Not seeing it at all over 5+ rolls at the right level → check for contest; if clear, it may just be variance, budget more; if contested, reassess.

Expected roll value

A roll is +EV when the expected board-strength gain (weighted by hit chance) exceeds the cost, where "cost" includes the interest you break and the levels you didn't buy. Practical heuristics that fall out of the math:

• Roll at the level where your target rarity is ≥25–30% per slot. Below that, the EV is almost always negative versus leveling first.
• The first ~10 gold of a roll-down is the most efficient — you're spending surplus above your interest cap. Past 50 → toward 30 you're paying real interest; past 30 → 0 you're all-in and need a specific justification.
• Stop-loss is part of the plan. "Roll to 20, then level regardless" is a bounded, calculable play. "One more roll" ten times is how stockpiles vanish for nothing.
• Rolling is worth more when it's cheap for the pool to give you what you want — right after a hoarder dies, right after you level into a new rarity band, when nobody contests your tier.