(I preface this with the warning that I can never keep the formulas for permutations, subluxations and combinations straight. Although I approve of using factorials in formulae as a way of making math more exciting.)

I think using a pair of dice as illustration is a different problem, since there are six ways to get a total of 7. But that is not the question here.

My example: a and b are running and six people vote. The final tally has to be one of the following:

a b
6 0
5 1
4 2
3 3
2 4
1 5
0 6

There are 7 possible vote totals, of which 1 is a tie. Probability of a tie is 1/7.

In a race between a and b as shown, I note that since the total number of votes is given a and b are not independent variables. Given the known total, once the value for a or b is known the other is also determined. By my thinking, if the race is only between a and b and 12,000 votes are cast, of the 12,000 possible outcomes (Wait. 12,001) one is a tie. so, by chance alone the probability of a tie score is 1/12,001.

Since there were actually six names on the ballot it gets messier, of course (Perhaps a Messier Number could be used here).

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