Consider an experiment that firstly involves rolling a fair four-sided die once and then secondly rolling a fair ten-sided die once. Let X denote the number rolled on the first roll and let Y denote the number rolled on the second roll.

(a) How many possible outcomes are there? That is, how many different observed values for (X, Y) can eventuate from this experiment? Explain how you got your answer.

(b) For the remainder of this question you may assume that the different observed values (outcomes) for (X, Y)are all equally likely to occur. What is the probability associated with each outcome occurring?

(c) Let A denote the event that the number rolled on the first roll is less than or equal to the number rolled on the second roll (i.e. XY). Calculate the probability of event A occurring. Show workings.

(d) What is the probability of A not occurring? Show workings.

(e) Now, let B denote the event that the number rolled on the second roll is exactly 3 times larger than the number rolled on the first roll

(i.e. Y= 3X). Calculate the probability of event B occurring. Show workings.

(f) Are events A and B mutually exclusive? Explain.

(g) What is the probability that event A occurs and event B occurs? Show workings.

(h) What is the probability that event A does not occur and event B occurs?. Show workings.

(i) What is the probability that either event A does not occur or event B occurs? Show workings.