Standard expected value question.
Collecting the full set
The question
A cereal brand hides one of six equally likely toys per box. How many boxes until you expect to own all six?
Have a go before you open the solution. If you are stuck after a few minutes, the first paragraph below is usually enough of a nudge.
Show the worked solution
The answer
14.7 boxes.
After you already hold k distinct toys, each box is new with probability (6-k)/6, so the expected wait for the next new one is 6/(6-k).
Sum for k from 0 to 5: 6 times (1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6).
The harmonic sum is 2.45, so the expected total is 14.7 boxes.
In general it is n times the nth harmonic number, which grows like n log n. The last item alone costs n boxes on average.
What it is testing
Decomposing a wait into independent geometric stages.
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