Math Says Yes
Concept

Expected Value

The long-run average outcome: each result weighted by how likely it is. It tells you whether a bet is favourable on average, but it ignores how badly a single rare outcome would hurt — which is why people rationally pay to avoid losses they could not absorb.

Lessons

An almost complete collection grid with one empty slot and many duplicates.
Completing a Set Takes Longer Than You Think
Filling in the last few items of a collection costs far more than the early ones, because duplicates pile up near the end.
A calm average marker beside a heavy rare-outcome tail.
The Average Isn't the Whole Story
Expected value ignores how badly the rare outcome would hurt.

Related Facts

Nature & Scale
4 min · easy
An almost-complete grid of collectible slots with a single empty slot and a pile of duplicates.
The last sticker costs the most
Completing a set of N collectibles takes far longer than N tries — the final few items hide behind a wall of duplicates, and the expected number of packs is about N·ln N.
Risk & Decisions
4 min · medium
A small figure safe under a big umbrella in a storm, with a few coins drifting away.
Insurance is a bet you want to lose
On average you pay in more than you ever get back. Buying it is still the smart move.

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