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

(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