Early days of a flavor hunt

Question

A big candy bin holds 6 flavors in equal proportion. You draw pieces one at a time (the bin is large enough that proportions stay effectively fixed). You have tasted just 1 flavor so far.

On average, how many more pieces until you taste a flavor you have not had yet?

Accepted Answer

Only the next new flavor matters. A single piece is new when it is one of the 6 - 1 = 5 flavors you have not tasted, with probability p = \frac{5}{6}. The geometric wait averages \frac{1}{p} = \frac{6}{5} = 1.2 pieces. Early on, almost every piece brings something new, so the wait is barely above one. Takeaway. The next-new-type wait \frac{n}{n-k} is smallest at the start (near 1) and only grows as your collection fills up.

Early days of a flavor hunt

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