I'm trying to work out the answer to this Q, and i'm stuck.
Assume that we have 100 voters and each voter has a two-thirds chance choosing correctly between binary options. We are conducting a binary vote between A and B. Further suppose that the difference in votes between the two positions is a 5% split: 55 voters choose A and 45 choose B. The voters are both independent and sincere.
How confident should we be, based on this distribution, that option A is correct?
If the voters are independent, given that $A$ is correct the chance of getting this vote is ${100 \choose 55}\frac {2^{55}}{3^{100}}$. Given that $B$ is correct the chance of getting this vote is ${100 \choose 45}\frac {2^{45}}{3^{100}}$. This gives the chance $A$ is correct as $\frac {1024}{1025}$ as the binomials are equal. If the voters are not independent we need to know about the correlations to be able to say anything.