Probability: is it raining? 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?  

Attempt:
To put it a bit more formally... Let $p_r$ be the prior probability for rain ($p_n=1-p_r$). Then the probability of rain given 3 "yes" replies $(y,y,y)$

$$P\{\text{rain}|y,y,y\}=\frac{P\{\text{rain}\bigcap (y,y,y)\}}{P\{y,y,y\}}$$
$$=\frac{P\{y,y,y|\text{rain}\}\cdot p_r}{P\{y,y,y|\text{rain}\}\cdot p_r+P\{y,y,y|\text{no rain}\}\cdot p_n}$$

Next, if we assume conditional independence of friends' replies the last formula becomes

$$=\frac{(2/3)^3\cdot p_r}{(2/3)^3\cdot p_r+(1/3)^3\cdot p_n}=\frac{p_r}{p_r+ p_n/8}$$

This answer is  duplicate from the following link
Facebook Question (Data Science)
Suppose it rains, then 3 yes answers would have been heard with probability $\frac{8}{27}$ if it does not rain, we'd hear those answers with probability $\frac{1}{27}$. We know we've heard them, so we're in one of two worlds, and one is 8 times more likely. So the intuitive answer is $\frac{8}{9}$, I'd say.