[math] \newcommand{\NA}{{\rm NA}} \newcommand{\mat}[1]{{\bf#1}} \newcommand{\exref}[1]{\ref{##1}} \newcommand{\secstoprocess}{\all} \newcommand{\NA}{{\rm NA}} \newcommand{\mathds}{\mathbb}[/math]

Assume that every time you buy a box of Wheaties, you receive

one of the pictures of the [math]n[/math] players on the New York Yankees. Over a period of time, you buy [math]m \geq n[/math] boxes of Wheaties.

• Use Theorem to show that the probability that you get all [math]n[/math] pictures is
  [[math]] \begin{eqnarray*} 1 &-& {n \choose 1} \left(\frac{n - 1}n\right)^m + {n \choose 2} \left(\frac{n - 2}n\right)^m - \cdots \\ &+& (-1)^{n - 1} {n \choose {n - 1}}\left(\frac 1n \right)^m. \end{eqnarray*} [[/math]]
  Hint: Let [math]E_k[/math] be the event that you do not get the [math]k[/math]th player's picture.
• Write a computer program to compute this probability. Use this program to find, for given [math]n[/math], the smallest value of [math]m[/math] which will give probability [math]\geq .5[/math] of getting all [math]n[/math] pictures. Consider [math]n = 50[/math], 100, and 150 and show that [math]m = n\log n + n \log 2[/math] is a good estimate for the number of boxes needed. (For a derivation of this estimate, see Feller.)

Notes