1- Assume we have two random variables X and Y. They have no conditional or unconditional dependencies. What is the numerical value of the following sentence? Justify your answer. [10 points]

i P (X=i | Y=y)

1- Assume M is an event which has a 50% chance (equivalent to 1/2) to happen. N is another event which may happen 33.33 % (equivalent to 1/3) of the times. We also know that the chance of M happening given that N does not happen is 25% (equivalent to 1/4). What is the chance of M happening given that N happens? Show your work. [20 points]

2- Consider the following Bayesian Network. What is the Joint probability of A does not happen and B happens, and C does not happen, and D happens? Show your work. [20 points]

3- Draw the resulting Bayesian network from the following sentence. [15 points]

P(Q) P(X | Q) P(Y) P(W | Q,X,Y) P(Z | X,W) P(G | Y,W,Z) P(M | W)

4- Two factories Factory A and Factory B design batteries to be used in mobile phones. Factory A produces 60% of all batteries, and Factory B produces the other 40%. 2% of Factory A's batteries have defects, and 4% of Factory B's batteries have defects. What is the probability that a battery is both made by Factory A and defective? Show your work. [10 points]

5- Consider the following Bayesian network. Which of the following sentences are true and which ones are False? Briefly justify your answers. [25 points]

A. Assuming we know there is rain, whether there is track maintenance does not affect the probability that the train is on time. B. Assuming we know there is track maintenance, whether or not there is rain does not affect the probability that the appointment is attended.

C. Assuming we know the train is on time, whether or not there is rain affects the probability that the appointment is attended. D. Assuming we know there is track maintenance, whether or not there is rain does not affect the probability that the train will be on time.

E. Assuming we know the train is on time, whether there is track maintenance does not affect the probability that the appointment is attended.