Imagine an election between two candidates.
A receives m votes, B receives n votes, and A wins (m>n).
If the ballots are cast one at a time, what is the probability that A will lead all the way throughout the voting process?
I'm guessing the formula is P(m,n) = (mn)/(m+n)
Obviously P(m,0)=1
and
P(m,1)=(m1)/(m+1) since B's lone vote cannot be first or second.
For P(m,2)
the denominator is C(m+2,2)=(m+1)(m+2)/2
the numerator requires the first two votes to be A = C(m,n) except the first four votes cannot go AABB which is only a single possibility so C(m,n)  1 = m(m1)/2  1 = (m+1)(m2)/2
and the fraction reduces to (m2)/(m+2)
From there things get complicated
Unallowed starts would be B, AB, AABB, AABABB, AAABBB. I didn't tabulate them by formula but by hand counting P(4,3)=5/35=1/7 which fits the pattern (43)/(4+3)
Edited on October 6, 2015, 12:59 pm

Posted by Jer
on 20151006 08:15:19 