@MISC{BALLOT_1.MIZ,
  AUTHOR = {P\k{a}k, Karol},
  TITLE = {Bertrand's Ballot Theorem},
  NOTE = {The paper has been financed by the
    resources of the Polish National Science Centre granted by decision
    no DEC-2012/07/N/ST6/02147.},
  SECTION1 = {Preliminaries},
  SECTION2 = {Properties of Elections},
  SECTION3 = {Properties of Dominated Elections},
  SECTION4 = {Main Theorem},
  DAY = {13},
  MONTH = {June},
  YEAR = {2014},
  ADDRESS1 = {Institute of Computer Science\\
             University of Bia{\l}ystok\\
	     Sosnowa 64, 15-887 Bia{\l}ystok\\Poland},
  SUMMARY = {In this article we formalize the Bertrand's Ballot Theorem
             based on \cite{BALLOT}. Suppose that in an election we have
             two candidates: $A$ that receives $n$ votes and $B$ that
             receives $k$ votes, and additionaly $n\geq k$. Then this
             theorem states that the probability of the situation where
             $A$ maintains more votes than $B$ throughout the counting
             of the ballots is equal to $(n-k)/(n+k)$.
             \par
             Bertrand's Ballot Theorem is listed \#30 at Freek Wiedijk's
             list of "Top 100 mathematical
             theorems" \cite{Freek-100-theorems}.}}


@ARTICLE{BALLOT,
  AUTHOR =       {M.~Renault},
  TITLE =        {Four Proofs of the Ballot Theorem},
  JOURNAL =      {Mathematics Magazine},
  YEAR =         {2007},
  volume =       {80},
  number =       {5},
  pages =        {345--352},
  month =        {December}
}






