Spin the Wheel

Seeing every name once takes about three times as many spins

A fair wheel with n segments needs, on average, n × (1 + ½ + ⅓ + … + 1/n) spins before every segment has come up at least once. That is a lot more than n:

Ten names need about 29 spins, not ten. And the multiplier is not fixed — it grows roughly like the logarithm of the size, so a fifty-segment wheel needs 4.5× rather than 2.9×. Bigger wheels are proportionally worse, not just absolutely worse.

This is the coupon collector problem, and it is the arithmetic behind every "why haven't I got that one yet" feeling in a sticker album, a loot box, or a randomised playlist.

And nearly all the waiting is for the last few

The cost is wildly uneven across the collection. With 10 segments, here is what each new distinct result costs on average:

The final segment alone accounts for 34% of the whole exercise — the first five together cost 6.5 spins, less than the last one does on its own.

That is the honest answer to a wheel that feels rigged. If you are waiting on one particular name, you are in the part of the process that genuinely takes longest, and a long wait is not evidence that anything is wrong.

It also points at when a wheel is the wrong tool. If everyone genuinely has to be picked once — a running order, a rota, assigning tasks — spinning is a slow way to get there and might never finish while you are watching. Shuffle the list and deal from it instead: that takes exactly 10 draws for 10 names, by construction, against about 29 spins. Use the wheel when each result should be independent, and a shuffle when the set has to be covered.