1 Exam III Review Math-132 (Sections 7.1, 7.2, 7.3, 7.4, 7.5, 7.6, 8.1, 8.2, 8.3) On this exam, questions may come from any of the following topic areas: - Union and intersection of sets - Complement of sets - Constructing and interpreting Venn Diagrams - Applying the additive rule to counts in sets - Applying the additive rule to probabilities of events - Applying the additive rule to word problems - The multiplicative principle - Tree diagrams - Simple probability calculations - Mutually exclusive and independent random outcomes - Probabilities of mutually exclusive and independent outcomes - Calculating probability given the odds - Calculating the odds given probability - Calculating the expected value - Calculating the expected value for the information in a word problem - Conditional probability of dependent events - Conditional probability of independent events - Probability calculation using the Baye s theorem Sections 7.1 #1) If { } { } what is: a) b) #2) If { } { } { } Find [ { } ] #3) If { } { } { } and { } Find [ { } ] #4) If { } { } { } Find [ { } ]
2 #5) If { } { } { } Find [ { } ] #6) A class of 28 students were surveyed and asked if they ever had dogs or cats for pets at home. 8 students said they only had a dog; 6 students said they only had a cat; 10 students said they had a dog and a cat; and 4 students said they never had a dog or a cat. Create a Venn Diagram that captures this information. [ ] 6 Cats 10 Dogs 8 4 #7) Out of 65 students, 28 are taking English Composition and 35 are taking Chemistry. 11 students are taking both classes. a) Draw the Venn Diagram for this information. [ answer: Do you see how this is done? Hint: Draw the Venn Diagram and place the intersection number, first] English U 17 11 Chemistry 24 13 b) How many are in neither class? [answer: 13]
3 c) How many are taking at least one of the classes? [answer: 52] d) What is the probability that a student is taking English but not Chemistry? [answer: 0.262] e) What is the probability that a student is taking both classes? [answer: 0.169] f) What is the probability that a student is taking English, given that the student is taking chemistry? [answer: 0.314] #8) In a group of 58 students, 24 are taking algebra, 12 are taking biology, 19 are taking chemistry, 7 are taking algebra and biology, 11 are taking algebra and chemistry, 6 are taking biology and chemistry, and 4 are taking all three courses. a) Draw the Venn Diagram for this information. [answer: Shown below. Do you see how this is done? Hint: Place the intersection number for all three courses first and work your way outward.] 10 A 3 4 7 2 3 B U 6 C 23 b) How many students are not taking any of these courses? [answer: 23 ] c) What is the probability that a student is not taking any of these courses? [answer: 0.397 ] d) What is the probability that a student is taking algebra and biology but not chemistry? [answer: 0.0517 ] e) What is the probability that a student is taking only biology? [answer: 0.0517] f) What is the probability that a student is taking exactly two of these courses? [answer: ]
4 Section 7.2 #9) Find ( ), given that ( ) and ( ) and ( ) #10) Find ( ), given that ( ) and ( ) and ( ) #11) Find ( ), given that ( ) and ( ) and ( ) #12) Motors, Inc. manufactured 250 cars with a GPS system, 205 with satellite system, and 70 with both these options. How many cars were manufactured if every car has at least one of these options? #13) In a survey of 600 business travelers, it was found that of two daily newspapers, New York times, and Wall Street Journal, 360 read New York Times, 112 read New York Times and Wall Street Journal, and 56 read only Wall Street Journal. a) How many read New York Times or Wall Street Journal? b) How many did not read either New York Times or Wall Street Journal? Section 7.3 #14) A man has 13 shirts and 5 ties. How many different shirts and tie arrangements can he wear? #15) A restaurant offers 3 different salads, 6 different main courses, 13 different desserts, and 7 different drinks. How many different launches are possible? #16) How many different ways can 4 people be seated in a row of 4 seats? #17) How many 4-letter code words are possible using the first 5 letters of the alphabet with no letters repeated? [ ] How many codes are possible when letters are allowed to repeat?
5 Section 7.4 #18) Three letters are picked from the alphabet (repetitions are allowed, assume order is important). Find the number of outcomes in the sample space. #19) A ball is picked from a box containing 8 orange, 5 red, 4 blue, and 9 green balls. Find the probability that a green or an orange ball is picked. #20) The 2008 income of U.S. families is described in the table: Income Level Number of families (in thousands) <$25,000 28,943 $25,000 - $49,999 29,178 $50,000 - $74,999 20,975 $75,000 - $99,999 13,944 24,022 What is the probability that a randomly selected family has an income of less than $50,000? Section 7.5 #21) If events E and F belong to the same sample space, and ( ), ( ), ( )=0.40 find ( ) #22) If events E and F belong to the same sample space, and ( ), ( ), ( )=0.80 find ( ) #23) If events A and B are mutually exclusive and if ( ), ( ), find the probability ( ). #24) Determine the probability of E if the odds against E are 4 to 13. #25) Determine the probability of the event E if odds against E are 5 to 2. #26) Determine the probability of E if the odds in favor of E are 3 to 1.
6 #27) Determine the probability of E if the odds in favor of E are 1 to 1. #28) Determine the odds for and against the event E if ( ). [ ] #29) Determine the odds for and against the event F if ( ) [ ] #30) Determine the odds for and against the event F if ( ) [ ] #31) Suppose events A and B are independent, with ( ) and ( ). a) Find the odds for A. b) Find the odds for (i.e., the odds against A) #32) Events A and B are mutually exclusive, with ( ) and ( ). a) Find the odds for A b) Find the odds for (i.e., the odds against A) #33) Anne is taking courses in both mathematics and English. She estimates her probability of passing mathematics at 0.3 and English at 0.4, and she estimates her probability of passing at least one of them at 0.53. What is her probability of passing both courses?
7 #34) In a survey of the number of TV sets in a house, the following probability table was constructed. Number of TV sets 0 1 2 3 4 or more Probability 0.03 0.26 0.33 0.19 0.19 Find the probability of a house having: a) 1 or 2 TV sets b) 1 or 2 or 3 TV sets c) At least 2 TV sets d) At most 3 TV sets #35) In 2009, there were roughly 352,000 trademark applications filed in the United States. From these applications, about 320,000 trademarks were issued. What are the odds that a trademark application will result in a trademark being issued? Hint: consider a ratio of trademark applications that received approval to those applications that were denied. #36) A financial consultant estimates that there is an 11% chance a mutual fund will outperform the market during any given year. She also estimates that there is a 10% chance that the mutual fund will outperform the market for the next two years. What is the probability that the mutual fund will outperform the market in at least one of the next two years? #37) Three letters, with repetition allowed, are selected from the alphabet. What is the probability that none is repeated? #38) What is the probability that, in a group of 7 people, at least two are born in the same month?
8 #39) A jar contains 6 white marbles, 2 yellow marbles, 2 red marbles, and 5 blue marbles. Two marbles are picked at random and without replacement. a) What is the probability that both are blue? b) What is the probability that exactly one is blue? c) What is the probability that at least one is blue? Section 7.6 #40) Attendance at a football game in a certain city results in the following pattern. If it is extremely cold, the attendance will be 40,000; if it is cold, it will be 50,000; if it is moderate, 70,000, and if it is warm, 90,000. If the probability for extremely cold, cold, moderate, and warm are 0.09, 0.41, 0.41, 0.09, respectively, how many fans are expected to attend each game? #41) A player rolls a fair six-sided die and receives a number of dollars equal to the number of dots appearing on the face of the die. What is the least the player should expect to pay in order to play the game? #42) In a raffle, 1000 tickets are being sold at $1.00 each. The first prize is $110 dollars, and there are 4 second prizes of $50 dollars each. By how much does the price of a ticket exceed its expected value? #43) A 20-year old man purchases a one-year life insurance policy worth $250,000. The insurance company determines that he will survive the policy period with probability 0.9986. a) If the premium for the policy is $470, what is the expected profit for the insurance company? b) At what value should the company set its premium so its expected profit will be $220?
9 #44) A bag contains 2 red balls and 4 green ones. In a game, customers are charged $0.68 to draw two balls from the bag one after another without replacing them back. For each red ball that a customer draws, he or she is paid $1. Assuming that the order of drawing each ball matters, by how much will a customer overpay to play this game? #45) In a lottery, 1000 tickets are sold at $0.32 each. There are 3 cash prizes: one for $130, one for $80, and one for $10. Alice buys 7 tickets. A) Considering the expected value of each ticket, what would have been a fair price for a ticket? B) In total, how much extra did Alice pay? Section 8.1 #46) If E and F are events with P(E)=0.2 P(F)=0.3 ( ) a) Find P(E F) b) Find P(F E) c) Find ( ) #47) If E and F are events with ( ) and P(E F)=0.5 what is P(F)=?
10 #48) Given the following tree diagram probabilities, find the probability of obtaining D in any final outcome. 0.87 C 0.11 A 0.13 D 0.89 B 0.28 C 0.72 D #49) If P(E)=0.71 P(F)=0.79 ( ) find ( ) #50) If P(E)=0.20 P(F)=0.65 ( ) P(E F)=? [ ( ) ] #51) A regular deck of cards has 52 cards. There are 4 suits of 13 cards each. The suits are called Clubs (black), Diamonds (red), Hearts (red), and Spades (black). Two cards are drawn at random and without replacement from the deck. What is the probability that the first card is red and the second is black? #52) Given the data in the following table, what is the probability that a customer likes the deodorant given he/she is from group I? Like the deodorant Did not like the deodorant No opinion Group I 167 65 24 Group II 100 85 13 Group III 48 63 9
11 #53) If the probability that a store runs out of raspberry lemonade Gatorade when it is on sale is 0.84, and the probability the Gatorade is on sale is 0.21, what is the probability that Gatorade is on sale and the store runs out of the stock if the two events are independent of each other? Section 8.2 #54) Let E and F be independent events, P(E)=0.11 P(F)=0.35. What is ( ) #55) If E and F are independent, and if P(E)=0.3 and ( ), find P(F)=? #56) If E and F are independent and P(E)=0.40, ( ), find P(F)=? #57) If E and F are independent, with P(E)=0.5, P(F)=0.39 a) Find P(E F) b) Find P(F E) c) Find ( ) d) Find ( ) #58) If E, F, and G are independent events and P(E)=2/3 P(F)=3/13 P(G)=2/17 find ( ) #59) If P(E)=0.5 P(F)=0.3 ( ) find P(E F)
12 #60) A box has 10 marbles in it, two of these are red, and 8 are white. Suppose we draw a marble from the box, replace it, and then draw another. Find the probability that a) Both marbles are red b) Just one is red #61) A marksman hits a target with probability 0.6. Assuming independence for successive firings, find the probabilities of the following events: a) One miss followed by two hits. b) Two misses and one hit (in any order). Section 8.3 #62) Events and are mutually exclusive and form a complete partition of a sample space with ( ), ( ). If E is an event in with ( ) ( ), compute ( ) #63) Events,, and are mutually exclusive and form a complete partition of a sample space with ( ), ( ) ( ). If E is an event in with ( ) ( ), and ( ) compute ( ) #64) Events and are mutually exclusive and form a complete partition of a sample space with ( ), and ( ). If E is an event in ( ) and ( ), compute ( ) and ( ) [ ( ) ( ) ] #65) Events,, and are mutually exclusive and form a complete partition of a sample space with ( ), ( ) ( ). If E is an event in and ( ), ( ) and ( ). Compute ( ) ( ) and ( ) [ ( ) ( ) ( ) ]