TITLE: Facebook Question (Data Science)
QUESTION [28 upvotes]: Out of curiosity, here's a question from Glassdoor (Facebook Data Science Interview)

You're about to get on a plane to Seattle. You want to know if you
  should bring an umbrella. You call 3 random friends of yours who live
  there and ask each independently if it's raining. Each of your friends
  has a 2/3 chance of telling you the truth and a 1/3 chance of messing
  with you by lying. All 3 friends tell you that "Yes" it is raining.
  What is the probability that it's actually raining in Seattle?

Using Bayesian analysis, it's pretty clear that you cannot solve this without the prior probability of raining in Seattle.
However, here's a different approach which I found interesting in the discussion contained in Glassdoor. 
Since all three friends said "Yes", the question basically boils down to what is the probability that all three friends are telling the truth given that all three friends said "Yes". Since you ask all three friends independently, the probability that all three friends are telling the truth is given by (2/3)(2/3)(2/3) = 8/27. Thus, the probability of it raining is 8/27.
While this is perhaps a bit compelling, I'm not sure if it's correct. Anyone have any ideas?
Thanks!

REPLY [2 votes]: d.k.o. provided a detailed answer for how to do the Bayesian analysis. I'll try and give an intuition for the second part of your question: 

Since all three friends said "Yes", the question basically boils down to what is the probability that all three friends are telling the truth given that all three friends said "Yes". Since you ask all three friends independently, the probability that all three friends are telling the truth is given by (2/3)(2/3)(2/3) = 8/27. Thus, the probability of it raining is 8/27.
While this is perhaps a bit compelling, I'm not sure if it's correct. Anyone have any ideas?

This approach is incorrect, precisely because it fails to take into account the a priori probability of rain in Seattle. To see that the probability of rain in Seattle is important, just reframe the question a bit: 
 - What if this question was posed about an imaginary location in which it never rains? 
 - Or what if it was about a location in which it always rains?
The approach you describe above obviously fails in both these cases: in the former case, your friends must be lying, and in the latter, they must be telling the truth. 
Bayesian analysis works in these "edge cases": in d.k.o.'s notation, if $p_r = 1$, then
$\mathsf{P}(\text{rain}\mid \{y,y,y\}) =1$, and similarly, if $p_r = 0$, then $\mathsf{P}(\text{rain}\mid \{y,y,y\}) =0$. 
(I suspect Robert Israel was trying to make a similar point about what happens if the prior probability of the event your friends are reporting on is $0$.)