Quiz 1 Comments

1. In problem 1, the sample space consists of ordered pairs, where the first coordinate denotes the result of the coin toss and the second coordinate the result of the die toss. So the answers to all five parts should be sets of ordered pairs. Also remember events are sets and sets are not numbers but collections of elements; the cardinality of a set is a number and the probability of an event is a number, but the set is not a number.

2. In problem 2, I neglected to give the probabilities of the simple events e₁, e₂, e₃, and e₄, as I intended. The problem can still be done without these values (see the solutions), but because of the confusion that this omission may have caused, everyone received full credit (4 points) for problem 2. However, be aware of the mistakes that some made:
       ·  the probability of a sample point is not 1/|S| unless the sample points are equally likely, and you were not told this. Some assumed it without saying so, which  is wrong, and some said that they were assuming it, which is also wrong (although less wrong).
       ·  in this problem, as in problem 1, some confused sets and elements with their cardinalities; once again, a set is not a number; a set is a collection of objects whose cardinality is a number.

3. I should not have asked part (c) of question 3, since I said that combinations should not be on the quiz; since the answer to part (d) depends on (c), I gave everyone full credit for 3(c)&(d) - 4 points; I did grade (a) and (b), both of which use only the multiplication rule. Combinations are needed in the answer to part (c) since the order of the garnishes is not important. To say that there are 12 × 11 choices for garnishes, as several did, says that the order in which the garnishes are selected is important. To say the there are 12² = 12 × 12 choices for garnishes, as a few did, says both that the order of the garnishes matters and that a garnish can be selected twice. Neither is allowed. Finally, the answer to part (d) uses the addition principle with the mutually exclusive (disjoint) cases of 0, 1, and 2 garnishes, so is the sum of the answers in parts (a), (b) and (c).

4. Since there were 11 parts to the questions, the maximum possible score was 22, effectively giving 2 points extra credit.