To get a feel for how Microsoft is thinking about AI, quite a few episodes of the Microsoft Research podcast are handy too: https://www.microsoft.com/en-us/research/blog/category/podca... .. if you really are going to interview at Microsoft, having an idea of what their researchers are doing might be a huge help.
Ignoring the P(Rain|Seattle), here's a way to look at this: let's say it's not raining. Then all three must be lying; the probability of that is 1/27. So the probability that it's raining, given that these three are saying it's raining, is 26/27
Either all three must be lying, or all three must be telling the truth, right? So wouldn't you have to discard the probabilities of them disagreeing? There would be a 1/27 chance of all lying, and wouldn't there only be an 8/27 chance of all telling the truth? The other 18/27 would be a mixture of lies and truths. So really, if we're just comparing the odds of those two outcomes to each other, would the probability just be 8/9 that it is raining in Seattle?
Been a while since I've done anything with probabilities, so perhaps I'm way off here.
You CAN'T ignore the extra information like P(Rain|Seattle) though. What if this were the question instead: "Three friends in Seattle told you that Earth exploded and everyone died. Each has a probability of 1/3 of lying. What’s the probability that the Earth exploded and everyone died?"
The correct answer. But the question supposes one can know with accuracy what the percent chance that a given statement is a lie, which imo is pretty ridiculous. Fine abstract mathematical problem, but not very useful
That doesn't work: if twenty of your friends told you it was raining, your method would tell you the probability was (2/3)^20 P(R), which is very small.
Your method was my first thought too - it took a while before I saw how to do it.
The replies to this don't look correct so here's my take:
P(R) = Probability of rain in Seattle
P(F) = Probability of three friends saying it's raining in Seattle
P(F) = P(F|R)*P(R) + P(F|R')*(1-P(R))
= (8/27)*P(R) + (1/27)*(1-P(R))
= (7/27)*P(R) + (1/27)
P(R|F) = (P(F|R)*P(R))/P(F)
= [(8/27)*P(R)] / [(7/27)*P(R) + (1/27)]
= 8*P(R) / (7*P(R) + 1)
So, for instance, if the true probability of rain in Seattle is 50%, the probability of it raining in Seattle given our three friends are saying that it is raining in Seattle is 4/4.5 = 89%
