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?

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?



-All say yes: all three lie or three say the truth
-P("allsaythetruth")=(23)3=827P("allsaythetruth")=(23)3=827
-P("alllie")=(13)3=127P("alllie")=(13)3=127
-P("allyes")=127+827=13P("allyes")=127+827=13
-Out of these numbers, there is 82713=8982713=89 chance it's actually raining

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