Consider the following problem:
Alice has decided to participate in a jury with three members, with the verdict decided by majority. To express her disinterest in the case, she decides to vote by flipping a fair coin. The other two members make the correct decision with probability $p \in (0, 1)$. How does this arrangement compare to a judge who makes the correct decision with probability $p$?
The probability the correct decision is made in the former scenario can be calculated as $\frac{p^2}{2} + \frac{p^2}{2} + 2\left(\frac{p(1-p)}{2}\right) = p$, so the probabilities are the same.
Because they are the same, I am wondering if there is a more elegant solution that argues by symmetry?
Imagine the judge on the jury. The probability that the judge’s correct decision is messed up by the other two and the probability that the judge’s incorrect decision is corrected by the other two are the same by symmetry, since they both require the two serious people to disagree and Alice to make one of her two arbitrary decisions.