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I didnt't math during the thinking pause, but my intuition was a second liar makes it worse (more likey to end up 50-50 situation) and additional liars make it better as you get to reduce noise.
Is there a scenario where the extra liar makes it worse, you would be better yelling lalalallala as they tell you the answer?
I paused and wrote out all the probabilities and saw no way to improve beyond 80% - I scrolled down hoping to be proven wrong!
(I'm the author) I think there's an annoying thing where by saying "hey, here's this neat problem, what's the answer" I've made you much more likely to actually get the answer! What I really wanted to do was transfer the experience of writing a simulation for a related problem, observing this result, assuming I had a bug in my code, and then being delighted when I did the math. But unfortunately I don't know how to t…
As I was writing the simulation I realized my error. I finished the simulation anyway, just because, and it has the expected 80% result on both of them.
My error: when we trust "both" we're also trusting Alice, which means that my case was exactly the same as just trusting Alice.
PS as I was writing the simulation I did a small sanity test of 9 rolls: I rolled heads 9 times in a row (so I tried it again with 100 million and it was a ~50-50 split). There goes my chance of winning the lottery!
So much of this breaks down when the binary nature of the variables involved becomes continuous or at least nonbinary.
It's an example of a more general interest of mine, how structural characteristics of an inferential scenario affect the value of information that is received.
I could also see this being relevant to diagnostic scenarios hypothetically.
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.
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)
P(A|AAAA) = p^4
P(A|BBBB) = (1-p)^4
PMF = (n choose k) * p^k * (1-p)^(n-k)Sailors in the past had a similar adage: “Never go to sea with two chronometers; take one or three.” They relied on precise clocks to calculate longitude.
But the chronometers are will sync with each other if you don't store them apart, which would result correlated noise that an average won't fix.
Instead of three independent signals, you'd evaluate: given how Alice and Bob usually interact, does their agreement/disagreement pattern here tell you something? (E.g., if they're habitual contrarians, their agreement is the signal, not their disagreement.)
Take it further: human + LLM collaboration, where you measure the ongoing conversational dynamics—tone shifts, productive vs. circular disagreement, what gets bypassed, how contradictions are handled. The quality of the collaborative process itself becomes your truth signal.
You're not just aggregating independent observations anymore; you're reading the substrate of the interaction. The conversational structure as diagnostic.
Earlier quoted context omitted.
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)
No, it's 50% -- given that e.g. the flip is H, the base probability is both 16% for HT and 16% for TH.
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.
Sailors in the past had a similar adage: “Never go to sea with two chronometers; take one or three.” They relied on precise clocks to calculate longitude.