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In problems 1 to 4, the event A1 = { the sum of a pair of 6-sided dice is 1 }, and A2, ..., A50 are defined in the same way (the number of the set is the sum). Similarly, B1 = { the product of a pair of 6-sided dice is 1 }, and B2, ..., B50 are defined in the same way.

Describe the event C = { the sum of a pair of dice is equal to their product } in terms of the events Ai and Bj.

Compute the event C from problem 1 as a set of ordered pairs of numbers. Explain the method you used to compute this set.

What is the probability of the event C? What assumptions do you need to make to compute this probability?

Are {A1, ..., A50} and {B1, ..., B50} *partitions* of U = { all
the ways two 6-sided dice can fall }? Explain why or why not. If
they are both partitions, are they the *same* partition?

Mr. X is a student in this class. Describe *three* different ways of
assessing the probability that Mr. X will *fail to attend* class on
June 15.

For each method, explain any assumptions you need to make to justify using the method.

- Using the definitions of
*independence*of events and*conditional probability*, show that*A*and*B*are independent if and only if*P[B mid A] = P[B]*and*P[A mid B] = P[A]*. - Give an example where
*P[A | B] = P[A]*, but*P[B | A] neq P[B]*. Are*A*and*B*independent?