Question

Three friends in Seattle each told you it’s rainy, and each person has a 1/3 probability of lying. What is the probability that Seattle is rainy? Assume the probability of rain on any given day in Seattle is 0.25.

Answer 1

Answer:

8/11 probability

Step-by-step explanation:

Bayesian stats: you should estimate the prior probability that it’s raining on any given day in Seattle. If you mention this or ask the interviewer will tell you to use 25%. Then it’s straight-forward:

P(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 )

**Bonus points if you notice that you don’t need a calculator since all the 27’s cancel out and you can multiply top and bottom by 4.

P(training | Yes,Yes,Yes) = 8 / ( 8 + 3 ) = 8/11