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Number Theory/Two candidates having same number of votes.


Dear Prof Vijilant

Is this question valid?

In a election out of n number of candidates,find out the probability that two candidates are having the same number of votes?

Is further information required as this could not be adequate for question in probability?


Hello Prashant
We need to know the number of votes cast. Suppose there are two candidates and 2v votes are made.  Assume candidate A can receive any number of votes from 0 to 2v with equal probability.  Then she has v votes with probability 1/(2v + 1).  With more than 2 candidates we are into the realm of partition theory.
Now with 3 candidates.  Imagine A's votes placed in a line, followed by a blank sheet of paper, then b's
votes, another blank sheet and then C's votes.  There are (2v + 2)C2 different arrangements =
(2v+2)(2v+1)/2 = (v+1)(2v + 1).  A and B will have the same number of votes if they each have 0,1,2,3...v votes, v+1 ways in all.  So the probability of A and B having equal votes is 1/(2v + 1).  It might be worth checking this out for a small value of v.  Since there are 3 pairs, this gives 3/(2v + 1).  But this is still not correct, since if v is a multiple of 3, they could all have the same number.  We would have counted this 3 times, so need to subtract 2 from 3(v+1) giving (3v + 1)/((v + 1)(2v + 1))
If there are 50 000 votes, that gives a probability of 1/100 001.  Very small.
Best wishes

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