SAMPLE ANSWERING FORMAT:

Of the 45 subsets of {0, 1, 2, 3, 4, 5, 6, 7, 8,9} with 2 elements, one is chosen uniformly at random. Let X be lesser element of this subset and let Y be the greater element of the subset. (i) Are the events "X = 4" and "Y =9" independent? [2] (ii) Are the random variables X and Y independent? [2] (iii) What is Pr(Y 3)? [3] [Full justification required.](3) One of the 256 subsets of {1, 2, 3, 4, 5,6, 7, 8} is chosen uniformly at random. Let X be the number of elements of this subset. Let Y be 0 if the subset is empty and be the least element of the subset otherwise. (i) Are the events "X = 1" and "Y = 3" independent? [2] (ii) Are the events "X = 2" and "Y =6" independent? [2] (iii) What is Pr(X = 2 | Y s = 12: because the event "X = 2 A Y = 6 means that the subset chosen had two elements and that its least element was 6, which means the subset was {6, 7} or {6, 8}. So the events "X = 2" and "Y = 6" are not independent because Pr( X = 2)Pr(Y = 6) # Pr(X = 2 AY =6) (iii) Pr(Y