# Math Exam 2 Review. NOTE: For reviews of the other sections on Exam 2, refer to the first page of WIR #4 and #5.

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1 Math 166 Spring 2007 c Heather Ramsey Page 1 Math Exam 2 Review NOTE: For reviews of the other sections on Exam 2, refer to the first page of WIR #4 and #5. Section Experiments, Sample Spaces, and Events An experiment is an activity with observable results (called outcomes). Sample Space - the set of all possible outcomes of an experiment Event - a subset of a sample space of an experiment An event E is said to occur in a trial of an experiment whenever E contains the observed outcome. Unions, intersections, and complements of events are found in the same ways as they are for sets. Two events E and F are mutually exclusive if E F = /0 (i.e., if it is impossible for both E and F to occur at the same time). Section Definition of Probability The probability of an event is a number between 0 and 1 inclusive that indicates the likelihood of that event occurring. The closer the probability is to 1, the more likely the event is to occur. Probability Distribution - a table that lists all of the simple events of an experiment and their corresponding probabilities. NOTE: The sum of all probabilities in a probability distribution is always 1. Uniform Sample Space - a sample space in which all outcomes are equally likely. If E = {s 1,s 2,...,s k } is an event of an experiment with sample space S, then P(E) = P(s 1 )+P(s 2 )+ +P(s k ). Section Rules of Probability Let S be a sample space of an experiment and suppose E and F are events of the experiment. Then 1. 0 P(E) 1 for any E. 2. P(S) = 1 3. If E and F are mutually exclusive, then P(E F) = P(E) + P(F). 4. If E and F are any two events of an experiment, then P(E F) = P(E) + P(F) P(E F). 5. P(E c ) = 1 P(E) (Rule of Complements) NOTE: When calculating probabilities, Venn Diagrams can sometimes be useful. De Morgan s Laws may also come in handy from time to time: (E F) c = E c F c and (E F) c = E c F c. Section Use of Counting Techniques in Probability Computing the Probability of an Event in a Uniform Sample Space - Let S be a uniform sample space and let E be any event. Then number o f f avorable outcomes in E P(E) = number o f possible outcomes in S = n(e) n(s)

2 Math 166 Spring 2007 c Heather Ramsey Page 2 1. Consider the propositions p: Bob will have a hamburger for lunch. q: Bob will have pizza for lunch. r: Fred will have a hamburger for lunch. (a) Write the proposition r (p q) in words. (b) Write the proposition (q p) r in words. (c) Write the proposition Bob and Fred will both have a hamburger for lunch, or Bob will have pizza for lunch, symbolically. (d) Write the proposition Bob will not have a hamburger or pizza for lunch, but Fred will have a hamburger for lunch, symbolically. 2. Write a truth table for each of the following. (a) ( p q) (p q)

3 Math 166 Spring 2007 c Heather Ramsey Page 3 (b) r (q p) Problems 3, 4, 5, 6, 7, and 10 are courtesy of Joe Kahlig. 3. True or False. U = {0,1,2,3,4,5,6,7,8,9} and A = {0,1,2,3,4,5} T F /0 A T F n(a) = 5 T F n({3, 4}) = 2 T F /0 A T F {1,3,5} A T F n(/0) = 1 T F {1,2,3} A T F 2 A T F 3 A C T F 2 A T F {/0} = /0 T F 0 = /0 4. A = {a,b,c} (a) List all subsets of A. (b) List all of the proper subsets of A. (c) Give an example of two subsets of A that are disjoint. If this is not possible, then explain why. 5. Shade the part of the Venn diagram that is represented by (a) (A C B) (C A)

4 Math 166 Spring 2007 c Heather Ramsey Page 4 (b) (B C) A C 6. Write down the set notation that would represent the shaded portion of the Venn diagram. A B A B C C 7. U = {0,1,2,3,4,5,6,7,8,9}, A = {1,3,5,7,9}, B = {1,2,4,7,8}, and C = {2,4,6,8}. Compute the following. a) (A B) C b) A C B c) A (B C) C 8. Any problem like numbers from section Let U be the set of all A&M students. Let (a) Describe the set (A D) F c in words. A = {x U x owns an automobile} D = {x U x lives in a dorm on campus} F = {x U x is a freshman} (b) Use set notation (,, c ) to write the set of all A&M students who are freshmen living on campus in a dorm but do not own an automobile.

6 Math 166 Spring 2007 c Heather Ramsey Page Find n(a B) if n(a) = 8, n(b) = 9, and n(a B) = An experiment consists of tossing a 4-sided die and flipping a coin. (a) Describe an appropriate sample space for this experiment. (b) Give two events of this sample space that are mutually exclusive. 13. A bag contains 3 pennies, a nickel, and two dimes. Two coins are selected at random from the bag and the monetary value of the coins (in cents) is recorded. (a) What is the sample space of this experiment? (b) Write the event E that the monetary value of the coins is less than 11 cents. (c) Write the event F that the nickel is drawn. (d) Are the events E and F mutually exclusive? Support your answer. (e) Write the event G that the value of the coins is more than 25 cents.

7 Math 166 Spring 2007 c Heather Ramsey Page Let S = {s 1,s 2,s 3,s 4,s 5,s 6 } be the sample space of an experiment with the following probability distribution: Outcome s 1 s 2 s 3 s 4 s 5 s Probability Let A = {s 1,s 3,s 5 }, B = {s 3,s 4,s 6 }, and C = {s 2,s 4 } be events of the experiment and suppose P(B) = (a) Fill in the missing probabilities in the probability distribution above. (b) Is this a uniform sample space? Why or why not? (c) Find each of the following: i. P(A) ii. P(C) iii. P(B c ) iv. P(A B) v. P(A B) (d) Are the events A and C mutually exclusive? Why or why not?

8 Math 166 Spring 2007 c Heather Ramsey Page Acme, Inc. advertised its products in two magazines: Magazine A and Magazine B. A survey of 400 customers revealed that 120 learned of its products from Magazine A, 95 learned of its products from Magazine B, and 70 learned of its products from both magazines. What is the probability that a person selected at random from this group saw Acme, Inc. s advertisement in (a) exactly one of these magazines? (b) at least one of these magazines? (c) in Magazine A but not Magazine B? Information about a Standard Deck of 52 Cards There are 4 suits: hearts, diamonds, clubs, and spades. Hearts and diamonds are red; clubs and spades are black. There are 13 cards in each suit: Ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, Jack, Queen, and King. Jacks, Queens, and Kings are called face cards, so there are 12 face cards in a standard deck of 52 cards. 16. One card is drawn at random from a standard deck of 52 playing cards. What is the probability that the card is (a) a club? (b) a face card? (c) a club or a face card? (d) neither a club nor a face card?

9 Math 166 Spring 2007 c Heather Ramsey Page Acme, Inc. ships lightbulbs in lots of 50. Before each lot is shipped, a sample of 8 lightbulbs is selected from the lot for testing. If any of the bulbs is defective, the entire lot is rejected. What is the probability that a lot containing 3 defective lightbulbs will still get shipped? 18. Is the following statement correct? The probability that Kurt spends less than \$15 on a new DVD is 0.4. Therefore the probability that Kurt spends more than \$15 on a new DVD is Let E and F be two events of an experiment with P(E) = 0.35, P(F) = 0.55, and P(E F c ) = (a) Find P(E F). (b) What is the probability that exactly one of these two events occurs? (c) Are E and F mutually exclusive? (d) Find the probability that at least one of the two events occurs people are selected at random. What is the probability that at least 2 of the people in this group were born in the same week? (There are 52 weeks in a year. Assume that all weeks are equally likely.)

10 Math 166 Spring 2007 c Heather Ramsey Page How many different 5-card poker hands are there when playing with a standard deck of 52 cards? 22. How many different 5-card poker hands are there that have (a) exactly 2 aces? (b) two pairs? (A pair is two cards with the same number or face on them.) 23. What is the probability that a 5-card poker hand (a) is a full house? (A full house is a hand that has 1 pair and 1 three-of-a-kind.) (b) is a flush? (A flush is hand in which all cards are of the same suit.) 24. How many different social security numbers are there if (a) there are no restrictions? (b) only the numbers 2, 3, 5, 7, and 8 are used, the first two numbers are not the same, and the last digit is odd?

11 Math 166 Spring 2007 c Heather Ramsey Page Rosa has 3 different red dresses, 6 different black dresses, 7 different blue dresses, and 2 different green dresses. (a) In how many ways can Rosa arrange these dresses in her closet if there are no restrictions? (b) In how many ways can Rosa arrange these dresses in her closet if dresses of the same color must be kept together? (c) Rosa has decided to pack 4 dresses for a trip. How many ways can she choose 4 dresses if i. there are no restrictions? ii. at most 1 of the dresses is blue? iii. exactly 1 is blue and exactly 2 are red? (d) If Rosa randomly selectes 6 dresses from her closet, what is the probability that i. exactly 4 are black or exactly 2 are green? ii. at least 1 is blue? 26. Referring to the previous problem, if dresses of the same color were identical, how many distinguishable arrangements of the dresses would be possible?

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