Let A=event that 1st friend said its rainy, B = event that 2nd friend said its rainy, C = event that 3rd friend said its rainy.

P(Rainy | A, B, C) = P(A,B,C|Rainy)*P(Rainy)/P(A,B,C) using Bayes Theorem

P(A,B,C)=P(A,B,C|Rainy)*P(Rainy) + P(A,B,C|Not Rainy)*P(Not Rainy)

Assuming P(Rainy)=P(Not Rainy)=0.5

P(A,B,C) = 0.5*((2/3)³+(1/3)³)

P(Rainy | A, B, C) = 0.5*(2/3)³/0.5*((2/3)³+(1/3)³)=8/9