A young baseball fan wants to collect a complete set of 262 baseball cards. The baseball cards are available in a completely random fashion, one per package of chewing gum.
The fan buys two packets of chewing gum each day. How long on average will it take the fan to get a complete set?
I understand that this is a coupon's collector's problem. So, the general formula for this is:
$$= (262)\ln_{262}$$
which gives me 1458.9 as an answer.
Now I divided this by two since he buys two packs a day. Thus, I get $729$ days.
What I am confused, is the $average$ part of the question. Does the average involve a different formula or calculation?
Almost, but the formula for the expectation should perhaps be half of that for the coupon collector's problem, so $$\frac12 \cdot 262 \sum_{n=1}^{262} \frac1n$$ and if you work this out it is closer to $805.3$ than $729$
The approximation based on the harmonic series would be something like $$\frac12 \cdot 262 \left( \log_e(262) + \gamma + \frac{1}{2\cdot262} + \ldots \right)$$ where $\gamma\approx 0.5772156649\ldots$ is the Euler–Mascheroni constant, making a substantial difference to the result