PROBABILITY AND STATISTICS - ANSWER THESE TWO QUESTIONS AND SHOW WORK

1. In a bag containing 7 red balls and 5 white balls, you first draw one ball from the bag, and return the ball you drew and two extra balls with the different color as the ball you drew into the bag. Then you draw another ball from the bag. (a) What is the probability that the second ball you drew is white? (b) Given that the second ball you drew is red, what is the conditional probability that the first ball you drew is white?

2. Roll two fair four-sided dice. Let X and Y be the die scores from the 1st die and the 2nd die, respectively, and define a random variable Z = X Y . (a). Find the pmf of Z. (b). Draw the histogram of the pmf of Z. (c). Find P{Z < 0}. (d). Are the events {Z < 0} and {Z is odd} independent? Why?