Homework #3: Probability

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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.

Problem 1

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

Problem 2

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

Problem 3

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

Problem 4

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?

Problem 5

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.

Problem 6

  1. 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].
  2. Give an example where P[A | B] = P[A], but P[B | A] neq P[B]. Are A and B independent?