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Voting Paradox

Stage: 4 and 5 Challenge Level: Challenge Level:2 Challenge Level:2

mccain
obama clinton Some relationships are transitive , such as `if A> B and B> C then it follows that A> C', but some relationships are intransitive , for example if A likes B and B likes C it does not necessarily follow that A likes C.

In a voting system, if A beats B and B beats C we might reasonably expect A to beat C.

In this example there are 3 candidates for election. The voters have to rank them in order of preference. Consider the case where 3 voters cast the following votes: ABC, BCA and CAB. In the sense that one candidate is preferred to another :

A beats B by 2 choices to 1.
B beats C by 2 choices to 1
but A loses to C, again by 2 choices to 1.

Three voters go to vote in this election and have to rank the candidates. First, check you agree that each voter has six possible ways in which they can do this.

Assuming the voters are just as likely to rank them in one order as another, what is the probability that they all vote in a way that results in a paradoxical (intransitive) outcome?