Comment by krackers

Although revisiting this, you have to be a bit careful about the argument.

Basically what you're doing is breaking down p(correct) = p(correct & agree) + p(correct & disagree) where former is 0.8*0.8 and latter is 0.8*0.2. Explicitly computing the conditional probability however makes calculating more difficult: p(correct | agree)*p(agree) + p(correct | disagree)*p(disagree). This is something like (16/17) * (0.8*0.8 + 0.2*0.2) + 0.5 * (0.8*0.2*2) which is not easy to arrive at intuitively unless you grind through the calculation.

So _conditioned_ on them agreeing you are right ~94% while conditioned on them disagreeing it's a coin-toss (because when they disagree exactly one is right, and it's equally likely to be alice or bob). Interesting case where the unconditional probability is actually more intuitive and easier than the conditional.