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. w

The question cut off, but assuming that the question is "What is the probability that they are all messing with you?" the answer is 1/27.

Each friend has a chance of 1/3 of messing with you, so if you multiply the chances of each friend together (1/3*1/3*1/3) you get the probability of all three of the events (your friends messing with you) happening.

(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)

The event "all three say yes" happens 1/3 = 8/27 + 1/27 of the time. Out of these times, there is an 88.9% = 8/9 = (8/27) / (1/3) chance that it is actually raining, and a 1/9 chance it is not raining