The equalization probability of the Polya urn
  We consider a Polya urn, started with b black and w white balls, where b>w. We compute the probability that there are ever the same number of black and white balls in the urn, and show that it is twice the probability of getting no more than w-1 heads in b+w-1 tosses of a fair coin. 
Timothy C. Wallstrom
math.PR math.ST stat.TH