How Many Single, Doubles, Triples..., 30, -copies
Should the Coupon Collector Expect with , 540, kinds of coupons
By Shalosh B. Ekhad
There are, 540, equiprobable baseball (or soccer) cards
placed at random in chewing gums. Suppose that a collector
keeps buying them until he (or she) has all the cards
The EXPECTED number of cards that he needs to buy is
3709.643640
When the collection is complete (of course it would be sooner or later!)
then, according to the Foata-Han-Lass formula, given a short proof
by Doron Zeilberger in
http://www.math.rutgers.edu/~zeilberg/mamarim/mamarimPDF/coupon.pdf ,
the collector should EXPECT to have
6.869710445, kinds of cards that he only has one copy of
17.54829232, kinds of cards that he has, 2, copies of
35.66005545, kinds of cards that he has, 3, copies of
55.46688834, kinds of cards that he has, 4, copies of
70.51456945, kinds of cards that he has, 5, copies of
76.40899983, kinds of cards that he has, 6, copies of
72.67809737, kinds of cards that he has, 7, copies of
62.02813843, kinds of cards that he has, 8, copies of
48.32314776, kinds of cards that he has, 9, copies of
34.84553868, kinds of cards that he has, 10, copies of
23.52908180, kinds of cards that he has, 11, copies of
15.02532759, kinds of cards that he has, 12, copies of
9.152148858, kinds of cards that he has, 13, copies of
5.357419294, kinds of cards that he has, 14, copies of
3.033761347, kinds of cards that he has, 15, copies of
1.671526607, kinds of cards that he has, 16, copies of
0.9006324407, kinds of cards that he has, 17, copies of
0.4766341216, kinds of cards that he has, 18, copies of
0.2486844991, kinds of cards that he has, 19, copies of
0.1283218949, kinds of cards that he has, 20, copies of
0.06565436620, kinds of cards that he has, 21, copies of
0.03337668684, kinds of cards that he has, 22, copies of
0.01688712758, kinds of cards that he has, 23, copies of
0.008514435553, kinds of cards that he has, 24, copies of
0.004282174945, kinds of cards that he has, 25, copies of
0.002149784969, kinds of cards that he has, 26, copies of
0.001077897290, kinds of cards that he has, 27, copies of
0.0005399792992, kinds of cards that he has, 28, copies of
0.0002703410855, kinds of cards that he has, 29, copies of
0.0001352898029, kinds of cards that he has, 30, copies of
WARNING: these are all expectations (averages), in real life
he (or she) may finish the collection sooner (or later),
and have different distributions of duplicates. But if there
are many collectors, the above numbers are the
averages among all these people.