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Number Theory/Two candidates having same number of votes.

Question
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?

Thanks
Prashant

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
vijilant
Questioner's Rating
 Rating(1-10) Knowledgeability = 10 Clarity of Response = 10 Politeness = 10 Comment Dear Prof Vijilant Thank you. Prashant

Number Theory

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Vijilant

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Most questions on number theory, divisibility, primes, Euclidean algorithm, Fermat`s theorem, Wilson`s theorem, factorisation, euclidean algorithm, diophantine equations, Chinese remainder theorem, group theory, congruences, continued fractions.

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Teacher of math for 53 years

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AQA Doncaster Bridge Club Danum Strings Orchestra Doncaster Conservative Club Danum Strings Orchestra Simply Voices Choir Doncaster TNS mystery shopping St Paul's Music Group Cantley

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Journal of mathematics and its applications M500 magazine

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BSc (Hons) Liverpool (Science). BA (Hons) OU (Mathematics)

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State Scholarship 1955 Highest Score in Yorkshire on OU course MST209 £50 prize First class honours in OU BA Mathematics

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I taught John Birt, former Director of the BBC in 1961. His homework book was the most perfect I have ever marked. And also the most neat. I could tell he was destined for great things. One of my classmates was the poet Roger McGough, and I have a mention in his autobiography.