baseball_cards.txt
Outline for the article "Baseball Card Collecting"
See http://datagenetics.com/blog/april32016/index.html

Abstract
  MISSING
Introduction
  Many fans of baseball collect baseball cards, hoping to amass a complete
  set.  
  Cards are sold in small wrapped randomly filled packets.
  Mathematics and computation give some insight into this problem.
The Baseball Card Collector's Problem
  Definition of the sample problem (50 distinct cards, buy cards one at a time)
  Consider task of collecting complete set
  Give example of turning over one playing card at a time, to "collect"
  one each of heart, club, diamond, spade.
  Give related examples: Happy Meal (6 different toys), cereal box (5 different)
Initial Analysis
  No matter how long we try, there is a chance of never getting a complete set.
  The first card is always a "keeper", the second card is almost surely a 
  keeper (49/50 chances it is distinct), similarly each new card has a
  decreasing chance of being new.
  At some point, it is more likely than not that the next card is a repeat.
  Thus, getting the last missing cards will be very difficult.
Mathematical Analysis
  Illustration: show a sample sequence of cards drawn;
  Demonstrate that, if you already have K-1 unique cards, the probability that
  the next card you draw is unique is PK=(N-(k-1))/N.
  The expected wait til the next unique card is CK=1/PK=N/(N-(k-1)).
  The expected total wait E(N) is the sum C1+C2+...+CN.
  E(N) = N * sum ( 1+1/2+1/3+...1/N)
  For our 50 card example, this gives E(50) = 224.95 and answers our main question.
Harmonic Numbers
  The sum for E(N) is N * H(N), where H(N) is called the harmonic sum.  
  Although it is natural to guess that H(N) tends to some limit,
  it blows up (very very slowly) as N goes to infinity.  
  For example, it is possible to build a "bridge" out of playing cards
  where the topmost card hangs out halfway, the card beneath hangs out
  1/3 of the way, and so on.  By using more and more cards, the bridge
  can (theoretically) reach any desired distance.
  As N increases, H(N) can be approximated by the natural logarithm plus
  a correction term: H(N) -> ln(n) + gamma
  Use this to approximate E(N).
Distribution of Drawings
  The expected value E(N) only gives us the average waiting time.
  If we actually do an experiment, we expect a different waiting time.
  If we do many experiments, we can see the distribution of waiting times.
  Show a graph of waiting times for the 50-card example.
  Note distinction between mean, mode, and median in the results.
Extensions of the Problem
  Suppose we have multiple collectors who can swap cards to improve their sets;
  Suppose some cards are more rare than others.  This is true for baseball cards.
Conclusion
  MISSING
References:
  MISSING