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Problem 1. In a certain lottery, 5 distinct numbers are consecutively chosen from the first 50 positive integers. a) If the selection order matters, how many possible outcomes are there? b) How many outcomes are there with the number 3 appearing second? c) If the order of the numbers is considered irrelevant, how many outcomes consist solely of even numbers? Problem 2. Let A, B, C be events. a) Use set notation to express the event that exactly one of A, B, or C occur. b) Use set notation to express the event that A occurs but all three do not. c) Suppose that |A| = 7, |C| = 5 and |A ? C| = 2. What is |A4C|? Problem 3. a) Suppose that P is a real-valued function defined on the subsets of the countable set ?. What properties must P satisfy in order to qualify as a probability? b) Using the axioms in part a), prove that E ? F implies P(E) ? P(F). c) Prove that for any events A, B ? F, P(A ? B) = P(A) + P(B) ? P(A ? B). (Hint: A ? B = (A \ B) t (A ? B) t (B \ A), A = (A ? B) t (A \ B), and B = (A ? B) t (B \ A).) Problem 4. We roll a fair six-sided die twice. a) Describe a sample space ? and a probability measure P to model this experiment. b) Let A be the event that the second roll is larger than the first. Compute P(A). Problem 5. In certain variants of poker, a hand is called a flush house if it consists of 3 cards of one suit and 2 cards of another suit. Given a perfectly shuffled deck of cards, what is the probability of being dealt a flush house? * There are 52 cards in a deck (13 ranks in 4 suits) and 5 cards in a hand. Each set of five cards is equally likely and any set satisfying the given conditions constitutes a flush house, regardless of whether it also happens to count as some other hand.

          Problem 1. In a certain lottery, 5 distinct numbers are
consecutively chosen from the first 50 positive integers.
a) If the selection order matters, how many possible outcomes
are there? b)  How many outcomes are there with the number 3
appearing second? c)  If the order of the numbers is
considered irrelevant, how many outcomes consist solely of even
numbers?
Problem 2. Let A, B, C be events.
a)  Use set notation to express the event that exactly one
of A, B, or C occur. b)  Use set notation to express the event
that A occurs but all three do not. c)  Suppose that |A| = 7,
|C| = 5 and |A ? C| = 2. What is |A4C|?
Problem 3. a)  Suppose that P is a real-valued function
defined on the subsets of the countable set ?. What properties must
P satisfy in order to qualify as a probability? b) Using the axioms
in part a), prove that E ? F implies P(E) ? P(F). c) Prove that for
any events A, B ? F, P(A ? B) = P(A) + P(B) ? P(A ? B). (Hint: A ?
B = (A \ B) t (A ? B) t (B \ A), A = (A ? B) t (A \ B), and B = (A
? B) t (B \ A).) 
Problem 4. We roll a fair six-sided die twice. a)  Describe
a sample space ? and a probability measure P to model this
experiment. b)  Let A be the event that the second roll is
larger than the first. Compute P(A). 
Problem 5. In certain variants of poker, a hand is called a
flush house if it consists of 3 cards of one suit and 2 cards of
another suit. Given a perfectly shuffled deck of cards, what is the
probability of being dealt a flush house? * There are 52 cards in a
deck (13 ranks in 4 suits) and 5 cards in a hand. Each set of five
cards is equally likely and any set satisfying the given conditions
constitutes a flush house, regardless of whether it also happens to
count as some other hand.

Added by Erica S.

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