- #1
Loren Booda
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Consider an election with only one Democrat, one Independent and one Republican candidate running. Assuming a random vote, what is the chance that all pairs of parties attain a majority?
g_edgar said:Depends on the number of voters. For example, with 1 voter, the probability is 0. In general, look up "binomial distribution" I guess: you are asking for the probability that some party attains at least half the votes.
In short the probability that no one candidate gets 50% or more of the vote.Loren Booda said:Thank you for the modification. Try answering what the probability is as the number of votes approaches infinity.
But not just "some" party of the three. I am asking what the chance is that all pairs of parties attain a majority, i.e., when the number of votes for all parties taken two at a time are greater than 50%.
bpet said:I ran a few Monte Carlo simulations ... which disagrees with every answer here so far.
bpet said:I ran a few Monte Carlo simulations and the probability seems to converge to between 0.085 and 0.09 as the number of voters increases, which disagrees with every answer here so far.
Consider the voter preferences, one of 6 equally-likely possibilities (ABC, ACB, BAC, BCA, CAB, CBA) and let N1,...,N6 be the number of voters with each, so we have a multinomial distribution. A has a majority preference over B if N1+N2+N5>N3+N4+N6, B beats C if N1+N3+N4>N2+N5+N6 and C beats A if N4+N5+N6>N1+N2+N3. We need to work out the probability of all three inequalities or all three reverse inequalities (for the case A<B<C<A) being satisfied simultaneously.
BTW this is an example of the Condorcet paradox and you're effectively looking for the probability that an election has no Condorcet winner.
To calculate the majority chances in a three-way election, you need to first determine the total number of votes cast. Then, you can divide the number of votes for each candidate by the total number of votes to determine their individual percentages. The candidate with the highest percentage of votes will have the best chance of winning the majority.
If there is a tie between two candidates in a three-way election, the third candidate will automatically have a lower chance of winning the majority. In this case, you can recalculate the majority chances by only considering the two tied candidates and their respective vote percentages.
In theory, it is possible for all three candidates to have an equal chance of winning the majority in a three-way election. However, this scenario is highly unlikely and would require an exact equal number of votes for each candidate.
Voter turnout and demographics can greatly impact the majority chances in a three-way election. For example, if one candidate has a strong support base among a particular demographic group, they may have a higher chance of winning the majority if that group has a high turnout. Additionally, a lower overall voter turnout can make it easier for a candidate to secure the majority with a smaller number of votes.
Yes, the majority chances can change as more votes are counted. This is especially true if there is a close race between the top two candidates, as even a small shift in vote percentages can greatly impact the majority chances. It is important to continue counting all votes in order to accurately determine the winner and their majority chances.