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So... we penalise bad answers, and, because {first box 0, second box} is particularly bad, it gets clobbered three times instead of just once, driving it negative. That's vague. But I'll outline a precise version.
Start with the consistent, YNN example, and ask the questions in turn. Before you start you are completely ignorant - probability ¼ of all boxes. After first question you narrow it to two boxes. View this as adding on a correction term:pic.twitter.com/HiqFHfcil8
You can add one of these correction terms for the other two questions as well, ending up with the same as before:pic.twitter.com/58Fzyh4IvC
And it also works for the inconsistent, NNN example! You can see how the bottom left box gets clobbered by the successive -¼s, as in my vague motivation at the start.pic.twitter.com/8mgW5CvKT2
Is this trick any use? I really don't know! But I definitely find this breakdown more illuminating than just plugging through the algebra.
Now the reason I'm interested in this is the link to quantum physics. This toy model is very similar to phase space for a qubit. In some sense it's slightly 'worse', as -½ is more negative than anything that comes up there. But the simple numbers make it easy to play with.
For more information on how all this relates to quantum physics, see my follow up post: https://drossbucket.wordpress.com/2019/08/01/negative-probability-now-with-added-equations/ … May try a thread version of that one, too, but that'll be harder work. This is enough for today!
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