
Expectations
John collects figures from cornflake packets. Each packet contains one figure, and http://alt2.mathlinks.ro/latexrender...00c274bdaa.gif 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:
http://alt2.mathlinks.ro/latexrender...7cb6fd36fc.gif
Here:
http://alt2.mathlinks.ro/latexrender...d34ab619e6.gif  number of packets required for collecting the second figure
http://alt2.mathlinks.ro/latexrender...4540b3dda3.gif  number of packets required for collecting the third figure, and so on.
Each http://alt1.mathlinks.ro/latexrender...da5414c0f2.gif has a geometric distribution:
http://alt2.mathlinks.ro/latexrender...00781f357d.gif
Hence:
http://alt2.mathlinks.ro/latexrender...61b912ce35.gif, and therefore:
http://alt1.mathlinks.ro/latexrender...70110830fc.gif
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.