(Solved): Problem 1 Consider a random experiment with sample space \( S=[0,1] \). We assume that the probabi ...

Problem 1 Consider a random experiment with sample space \( S=[0,1] \). We assume that the probability of an event of the form \( C:=[a, b] \) corresponding to the interval between any two numbers \( a, b \) satisfying \( 0 \leq a \leq b \leq 1 \) is given by \( P(C)=b-a \). We also define the events \( A:=[0.1,0.6] \) and \( B:=[0.4,0.7] \) (a) Compute \( P(A \mid B) \) and \( P\left(A \mid B^{C}\right) \). (b) Based on your answer to (a), are \( A \) and \( B \) independent? (c) Use your answer to (a) and the law of total probability to obtain \( P(A) \). (d) Use your answers to (a) and (c) and Bayes Law to calculate the conditional probability \( P(B \mid A) \). (e) For an event \( C:=[a, b] \) derive a pair of values \( a, b, 0