I am trying to make better user experience for my poll application and one of ideas I have for this is breaking multi choices poll to binary polls.
Is statistical result of a question with these 4 options (a or b or c or d) equals with sum of statistical results of these 6 binary questions?:
(a or b), (a or c), (a or d), (b or c), (b or d), (c or d)
assume I ask these questions from separate people and I ask these questions equally
for example: assume we have a society with 1000 members if we ask all of them the first question, people will answer it with this distribution: a:60%, d:20%, b:15%, c:5%
now assume if we ask those six binary questions from all members of that society and then we sum each vote of winners of each of those questions.
we ask 1000 times a or b and answer is a:600,b:400
we ask 1000 times a or c and answer is a:500,c:500
we ask 1000 times a or d and answer is a:100,d:900 ....)
we sum number of votes on a and it is 1300 =600+500+100
we do similar thing with b,c and d, is this result similar with result of first question?(a:60%, d:20%, b:15%, c:5% ) and is meaning of this result similar with first one? ===update=== @Bruno mention to inconsistency of result of this approach and I think it can be solved if we do not show all combination to all voters. If voter1 chooses "a" between "a" and "b" we do not show questions with any combination of "b".