P(r | y, y, y) = P(y,y,y | r) * P(r) / (P(y,y,y | r)*P(r) + P(y,y,y | not r)*(1-P(r))
Let's write P(r) = p (probability that it is raining)
Moreover, we have P(y,y,y|r) = P(y|r)^3 = (2/3)^3 (Truth hypothesis + Bayes and independence)
and P(y,y,y|not r) = (1/3)^3 (same)
Then, plugging it into the bayes' formula, we have.--
P(r | y,y,y) = 8 p / (7p + 1)
Note that if p = 0 (say Seattle is in the middle of the Sahara, the probability of raining is always zero, independently from the three answers)