3. 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?

) 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. X Y ). 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.