[gweinberg]

Bob isn't giving you any actionable information. If Alice and Bob agree, you're more confident than you were before, but you're still going to be trusting Alice. If they disagree you're down to 50% confidence, but you still might as well trust Alice.

[titanomachy]

Better than 50% confidence: they only lie 20% of the time, so when they disagree it's still 64% likely to be heads (.8 x .8)

[npinsker]

No, it's 50% -- given that e.g. the flip is H, the base probability is both 16% for HT and 16% for TH.

[maxbond]

To complete the circle, now that we have winnowed the space down to these options, we would normalize them and end up with 0.16 / (0.16 + 0.16) = 0.5 = 50% in both cases.

The reason I'm not putting % signs on there is that, until we normalize, those are measures and not probabilities. What that means is that an events which has a 16% chance of happening in the entire universe of possibility has a "area" or "volume" (the strictly correct term being measure) of 0.16. Once we zoom in to a smaller subset of events, it no longer has a probability of 16% but the measure remains unchanged.

In this previous comment I gave a longer explanation of the intuition behind measure theory and linked to some resources on YouTube.

https://news.ycombinator.com/item?id=35796740