John collects figures from cornflake packets. Each packet contains one figure, and distinct figures make a complete set. Find the expected number of packets John needs to buy to collect a complete set.

Solution

The required number of packets is:



Here:

- number of packets required for collecting the second figure
- number of packets required for collecting the third figure, and so on.

Each has a geometric distribution:



Hence:

, and therefore:




Now the question arrises:

How much easier is it to collect half the set than the complete set? In particular, find the expected number of packets John needs to buy to collect half the set.

I am stuck on this question, despite it looks quite similar to the first.