I thought about this a little differently from a non-bayes perspective.

It's raining if any ONE of the friends is telling the truth, because if they are telling the truth then it is raining. If all of them are lieing, then it isn't raining because they told you that it was raining.

So what you want is the probability that any one person is telling the truth.

Which is simply 1-Pr(all lie) = 26/27

Anyone let me know if I'm wrong here!

Bayesian stats: you should estimate the prior probability that it's raining on any given day in Seattle. If you mention this or ask the interviewer will tell you to use 25%. Then it's straight-forward:

P(raining | Yes,Yes,Yes) = Prior(raining) * P(Yes,Yes,Yes | raining) / P(Yes, Yes, Yes)

P(Yes,Yes,Yes) = P(raining) * P(Yes,Yes,Yes | raining) + P(not-raining) * P(Yes,Yes,Yes | not-raining) = 0.25*(2/3)^3 + 0.75*(1/3)^3 = 0.25*(8/27) + 0.75*(1/27)

P(raining | Yes,Yes,Yes) = 0.25*(8/27) / ( 0.25*8/27 + 0.75*1/27 )

**Bonus points if you notice that you don't need a calculator since all the 27's cancel out and you can multiply top and bottom by 4.

P(training | Yes,Yes,Yes) = 8 / ( 8 + 3 ) = 8/11

But honestly, you're going to Seattle, so the answer should always be: "YES, I'm bringing an umbrella!"

(yeah yeah, unless your friends mess with you ALL the time ;)