Understand conditional probability.
AP Statistics Objectives Ch15 Understand conditional probability. Understand concept of independence. Know how and when to apply General Addition Rule. Know how and when to apply General Multiplication Rule. Know how to find probabilities for compound events
AP Statistics Objectives Ch15 Know how to find probabilities for compound events Know how to make and use a tree diagram to understand conditional probabilities and reverse conditioning. Conditional Probability Independence Tree Diagram
Vocabulary Sample Space Conditional Probability Independence Tree Diagram Chapter 15 Notes Chp 15 Practice Extra Practice
Chapter 15 Assignment Chapter 14 Assignment Part 1 Answers Chapter 14 Assignment Part 2 Answers If two events A & B are disjoint, P(A OR B) = P(A) + P(B)
Addition Rule The Addition Rule only works if the two events are disjoint. If two events A & B are disjoint, P(A OR B) = P(A) + P(B) A B If two events A & B are disjoint, P(A OR B) = P(A) + P(B)
1. Addition Rule If two events A & B are disjoint, P(A OR B) = P(A) + P(B) A B 2. General Addition Rule For any two events A & B, P(A OR B) = P(A) + P(B) P(A & B) A B Suppose that P(H) = .25, P(N) = .18, P(T) = .14.
Ex 1) A large auto center sells cars made by many different manufacturers.Three of these are Honda, Nissan, and Toyota. Suppose that P(H) = .25, P(N) = .18, P(T) = .14. a. Are these disjoint events? Yes, the cars only have one manufacturer. b. P(H or N or T) = = .57 c. P(not (H or N or T)) = = .43 Ex. 2) Musical styles other than rock and pop are becoming more popular.A survey of college students finds that the probability they like country music is .40. The probability that they liked jazz is .30 and that they liked both is What is the probability that they like country or jazz? P(C or J) = P(C) + P(J) P(C &J) = .4 + .3 .1 = .6
Ex. 2) Musical styles other than rock and pop are becoming more popular.A survey of college students finds that the probability they like country music is .40. The probability that they liked jazz is .30 and that they liked both is What is the probability that they like country or jazz? a. Are these events disjoint? They are not disjoint events. The students can like both music types. b. What is the probability that they like country or jazz? C J .3 .1 .2 P(C or J) = P(C) + P(J) P(C &J) = .1 = .6 3. Independent Two events are independent if knowing that one will occur (or has occurred) does not change the probability that the other occurs A randomly selected student is female. What is the probability she plays soccer for SHS? What is the probability she plays football for SHS? Independent, P(student plays soccer) = P(stud. plays soccer given they are female). Not Independent, P(student plays ftball) P(stud. plays ftball given they are female) 4. The Multiplication Rule only works if the two events are independent of each other.
P(A & B) = P(A) P(B) P(A & B) = P(A) P(B|A) 5. General Multiplication Rule:
4. The Multiplication Rule only works if the two events are independent of each other. P(A & B) = P(A) P(B) 5. General Multiplication Rule: For any two events A & B, P(A & B) = P(A) P(B|A) P(B|A) is read The probability of B given A. Ex. 3) A certain brand of light bulbs are defective five percent of the time. You randomly pick a package of two such bulbs off the shelf of a store. a. Can you assume they are independent? b. What is the probability that both bulbs are defective? Yes, since they are randomly selected. P(Defect & Defect) = = (0.05)(0.05) = = 0.0025 (0.05)(0.05) 2nd Bulb good 0.95 1st Bulb good 0.95 defective
Ex. 3) A certain brand of light bulbs are defective five percent of the time. You randomly pick a package of two such bulbs off the shelf of a store. a. Can you assume they are independent? b. What is the probability that both bulbs are defective? Yes, since they are randomly selected. 0.05 0.95 2nd Bulb good defective 0.05 0.95 1st Bulb good defective (0.05)(0.05) = P(exactly one D) = P(D & DC) or P(DC & D) = (.05)(.95) + (.95)(.05)
Ex. 3) A certain brand of light bulbs are defective five percent of the time. You randomly pick a package of two such bulbs off the shelf of a store. c. What is the probability that exactly one bulb is defective? P(exactly one D) = P(D & DC) or P(DC & D) = (.05)(.95) + (.95)(.05) = 0.095 Ex. 3) A certain brand of light bulbs are defective five percent of the time. You randomly pick a package of two such bulbs off the shelf of a store. c. What is the probability that exactly one bulb is defective? 0.05 0.95 2nd Bulb good defective 0.05 0.95 1st Bulb good defective (.95)(.05) = 0.475 + 0.475 (.05)(.95) = 0.095 d) What is the probability that at least one bulb is defective?
Ex. 3) A certain brand of light bulbs are defective five percent of the time. You randomly pick a package of two such bulbs off the shelf of a store. d) What is the probability that at least one bulb is defective? P(at least one D) = P(D & DC) or P(DC & D) or (D & D) = (.05)(.95) + (.95)(.05) + (.05)(.05) = .0975 At Least One The probability that at least oneoutcome happens is 1 minus the probability that no outcomes happen. P(at least 1) = 1 P(none) P(at least one D) = 1 P(DC & DC) = 1 (.95*.95) = .0975
Ex. 3 revisited) A certain brand of light bulbs are defective five percent of the time. You randomly pick a package of two such bulbs off the shelf of a store. d. What is the probability that at least one bulb is defective? P(at least one D) = 1 P(DC & DC) =1 (.95*.95) = Ex. 3) A certain brand of light bulbs are defective five percent of the time. You randomly pick a package of two such bulbs off the shelf of a store. d) What is the probability that at least one bulb is defective? 2nd Bulb good 0.95 1st Bulb good 2nd Bulb defective 0.95 (.95)(.05) = 0.0475 0.05 1st Bulb defective 2nd Bulb good 0.95 (.05)(.95) = + 0.0475 0.05 2nd Bulb defective (.05)(.05) = + 0.0025 0.05 0.0975 Example 4 If P(A) = 0.45, P(B) = 0.35, andA & B are independent. Find P(A or B). a. Is A & B disjoint? NO, independent events cannot be disjoint So. P(A or B) = P(A) + P(B) P(A & B) b. How do we find P(A & B)? Because they are independent, we multiply. So. P(A or B) = P(A) + P(B) P(A)(B) P(A or B) = (.35) = 0.6425 Example 4 : If P(A) = 0. 45, P(B) = 0. 35, and A & B are independent
Example 4 :If P(A) = 0.45, P(B) = 0.35, andA & B are independent. Find P(A or B). P(A&B) = P(A)*P(B) B A = (0.45)(0.35) 0.2925 0.1925 = 0.1575 0.2925 0.45 = 0.6425 0.35 = Example 6 If P(A) = 0.45, P(B) = 0.35, andA & B are independent. Find P(A or B). A NOTA B 0.35 NOT B 0.45 0.1575 0.65 1 0.35 = 0.65 0.55 1.00 1 0.45 = 0.55 P(A&B) = P(A)*P(B) = (0.45)(0.35) = Ex 5) Suppose I will pick two cards from a standard deck without replacement.What is the probability that I select two spades? a. Are the cards independent? NO, once I select 1 card the probability changes. P(1st Spade) = 13/52 P(2nd Spade | Spade 1st) = 12/51 Probability of 2nd Spade givenSpade 1st b. P(Spade & Spade) = P(1st Spade) P(2nd Spade | Spade 1st ) = 13/52 12/51= 0.059 Ex 6) For a sales promotion the manufacturer places winning symbols under the caps of 10% of all Dr. Pepper bottles.You buy a six-pack.What is the probability that you win something? Dr. Pepper P(at least one winning symbol) = 1 P(no winning symbols) = .4686 6. Conditional Probability
A probability that takes into account a given condition P(B|A) =() () = P(Sport | Female) = ( ) ()
Ex 7) a.In a recent study it was found that the probability that a randomly selected student is a girl is .51 and is a girl and plays sports is If the student is female, what is the probability that she plays sports? P(Sport | Female) =( ) () =. . = .1961 . . = .1961 Sport NO Sport Female Male 0.10 0.41 0.51 0.49 1.00
Ex 7) a. In a recent study it was found that the probability that a randomly selected student is a girl is .51 and is a girl and plays sports is If the student is female, what is the probability that she plays sports? 0.51 0.10 = 0.41 Sport NO Sport Female Male 0.10 0.41 0.51 0.49 1 0.51 = 0.49 1.00 . . = .1961 .31 = . .1519 = x P(Sport | Male) = ( ) ()
Ex 7) b. The probability that a randomly selected student plays sports if they are male is What is the probability that the student is male and plays sports if the probability that they are male is .49? P(Sport | Male) =( ) () .31 = . =x x = .1519 . = . Sport NO Sport Female Male 0.10 0.41 0.51 0.49
Ex 7) b. The probability that a randomly selected student plays sports if they are male is What is the probability that the student is male and plays sports if the probability that they are male is .49? Sport NO Sport Female Male 0.10 0.41 0.51 0.49 1.00 . =. x = .1519 Sport NO Sport Female Male 0.41 0.51 0.49 1.00
OK. Lets finish the table Sport NO Sport Female Male 0.10 0.41 0.51 0.1519 0.3381 0.49 0.2519 0.7481 1.00 0.49 = = = P(Female | Sports) = ( ) ()
c. Whats the probability that someone who plays sports is female? Sport NO Sport Female Male 0.10 0.41 0.51 0.1519 0.3381 0.49 0.2519 0.7481 1.00 P(Female | Sports) =( ) () =. . = .397 Probabilities from two way tables
Example 8 Probabilities from two way tables StuStaffTotal American European Asian Total a. What is the probability that the driver is a student? .% Probabilities from two way tables
Example 8 Probabilities from two way tables StuStaffTotal American European Asian Total b. What is the probability that the driver drives a European car? .% Probabilities from two way tables
Disjoint so:P(Am or Asian) = P(Am) + P(Asian) Probabilities from two way tables StuStaffTotal American European Asian Total c. What is the probability that the driver drives an American or Asian car? Disjoint .% Probabilities from two way tables
Not Disjoint so: P(Staff or Asian) = P(Staff) + P(Asian) P(Staff & Asian) Probabilities from two way tables StuStaffTotal American European Asian Total d. What is the probability that the driver is staff or drives an Asian car? Not Disjoint = 47 359 = % Probabilities from two way tables
Joint Distribution: P(Staff and Asian Car) Probabilities from two way tables StuStaffTotal American European Asian Total e. What is the probability that the driver is staff and drives an Asian car? = .% Probabilities from two way tables
Conditional Distribution: P(Am Car|Student) Probabilities from two way tables StuStaffTotal American European Asian Total f.If the driver is a student, what is the probability that they drive an American car? | = .% Probabilities from two way tables
Conditional Distribution: P(Student|Euro Car) Probabilities from two way tables StuStaffTotal American European Asian Total g. What is the probability that the driver is a student if the driver drives a European car? | = .% Time for Final Word on INDEPENDENCE
7. We know that two events are independent if the probability of one doesnt change just because of the other event. If Event A and Event B are Independent, then
= () (|) = P(A) = P(A|B) So if then Events A and B are independent Conditional Probability
StuStaffTotal American European Asian Total 8. Parts of a Tree Diagram P(B|A) P(A)*P(B|A) Conditional Probability Joint Probability Probability American & Student 107/359 P(A) Student 107/212 50.5% 29.81% Staff 105/212 American & Staff 105/359 American 212/359 49.5% 29.25% 59.1% European & Student 33/359 Student 33/45 European 45/359 73.3% 9.19% Staff 12/45 12.5% European & Staff 12/359 26.7% Asian 102/359 3.34% Student 55/102 Asian & Student 55/359 28.4% 53.9% 15.32% Staff 47/102 Asian & Staff 47/359 46.1% 13.09% 9. Draw with Replacement Blue & Blue Blue 50.0% 25.0% Green Blue
Green & Green 25.0% 10. Draw without Replacement
Blue & Blue Blue 33.3% 17% Green Blue & Green Blue 66.7% 50.0% 33% Green & Blue Blue Green 66.7% 33% 50.0% Green 33.3% Green & Green 17% Example 9: Management has determined that customers return 12% of the items assembled by inexperienced employees, whereas only 3% of the items assembled by experienced employees are returned. Due to turnover and absenteeism at an assembly plant, inexperienced employees assemble 20% of the items.Construct a tree diagram or a chart for this data. What is the probability that an item is returned? If an item is returned, what is the probability that an inexperienced employee assembled it? Example 9: a. Construct a tree diagram.
Management has determined that customers return 12% of the items assembled by inexperienced employees, whereas only 3% of the items assembled by experienced employees are returned. Due to turnover and absenteeism at an assembly plant, inexperienced employees assemble 20% of the items.a. Construct a tree diagram for this data. P(Inexp. AND Returned) = = .20 * .12 = .024 = 2.4% P(Inexp. AND Not Returned) = = .20 * .88 = .176 = 17.6% P(Exp. AND Returned) = = .80 * .03 = .024 = 2.4% P(Exp. AND Not Returned) = = .80 * .97 = .776 = 77.6% Example 9: Management has determined that customers return 12% of the items assembled by inexperienced employees, whereas only 3% of the items assembled by experienced employees are returned. Due to turnover and absenteeism at an assembly plant, inexperienced employees assemble 20% of the items.Construct a tree diagram or a chart for this data. b. What is the probability that an item is returned? P(Return) = P(Return & Inexp.) + P(Return & Exp.) = 2.4% + 2.4%= 4.8% Example 9: Management has determined that customers return 12% of the items assembled by inexperienced employees, whereas only 3% of the items assembled by experienced employees are returned. Due to turnover and absenteeism at an assembly plant, inexperienced employees assemble 20% of the items.Construct a tree diagram or a chart for this data. c. If an item is returned, what is the probability that an inexperienced employee assembled it? P(Inexp. | Returned) = Example 9: P(Returned | Inexp.) = .12 Not same as P(Inexp. | Returned)
Management has determined that customers return 12% of the items assembled by inexperienced employees, whereas only 3% of the items assembled by experienced employees are returned. Due to turnover and absenteeism at an assembly plant, inexperienced employees assemble 20% of the items.. P(Returned | Inexp.) = .12 Not same as P(Inexp. | Returned) P(Not Returned | Inexp.) = .88 P(Returned | Exp.) = .03 P(Not Returned | Exp.) = .97 (. ) () P(Inexp. | Returned) =
Example 9: Management has determined that customers return 12% of the items assembled by inexperienced employees, whereas only 3% of the items assembled by experienced employees are returned. Due to turnover and absenteeism at an assembly plant, inexperienced employees assemble 20% of the items.Construct a tree diagram or a chart for this data. c. If an item is returned, what is the probability that an inexperienced employee assembled it? (. )() P(Inexp. | Returned) = =..= .5 = 50% Example 10 Dr. Carey has two bottles of sample pills on his desk for the treatment of arthritic pain.He often grabs a bottle without looking and takes the medicine.Since the first bottle is closer to him, the chances of grabbing it are He knows the medicine from this bottle relieves the pain 70% of the time while the medicine in the second bottle relieves the pain 90% of the time.What is the probability that Dr. Carey grabbed the first bottle given his pain was not relieved? Example 10 relieved not .7 1st .6 .3 relieved not .9 .4 2nd .1 =
Example 10 = =. relieved .7 1st .6 .3 = (.6)(.3) = 0.18 not relieved not .9 .4 2nd .1 =
Example 10 = =. =. . .2 relieved .7 1st .6 .3 = (.6)(.3) = 0.18 not relieved = = 0.22 .9 .4 2nd = (.4)(.1) = 0.04 .1 not =
Example 10 = =. =. . .2 relieved .7 1st .6 .3 = (.6)(.3) = 0.18 not relieved = = 0.22 .9 .4 2nd = (.4)(.1) = 0.04 .1 not Chp 15 p.362 #3 Real estate ads suggest that 64% of homes for sale have garages, 21% have swimming pools, and 17% have both features. What is the probability that a home for sale has: a) a pool or a garage? NOT DISJOINT P(Garage U Pool) = P(G) + P(Pool) P(G Pool) = = 68% Garage Pool 64% - 17% = 47% 47% 17% 4% 21% - 17% = 4% P(Garage U Pool) = = 68% Chp 15 p.362 #3 P(NOT(Garage U Pool)) = P((Garage U Pool)C)
Real estate ads suggest that 64% of homes for sale have garages, 21% have swimming pools, and 17% have both features. What is the probability that a home for sale has: b) Neither a pool nor a garage? P(NOT(Garage U Pool)) = P((Garage U Pool)C) = 1 P(Garage U Pool) = 1 .68 = 32% Chp 15 p.362 #3 P(Pool Garagec) = 4% c) a pool but no garage?
Real estate ads suggest that 64% of homes for sale have garages, 21% have swimming pools, and 17% have both features. What is the probability that a home for sale has: c) a pool but no garage? P(Pool Garagec) = 4% Chp 15 p.363 #5 A check of dorm rooms on a large college campus revealed that 38% had refrigerators, 52% had TVs, and 21% had both a TV and a refrigerator. Whats the probability that a randomly selected dorm room has : a) A TV but no refrigerator? TV Frig 52% - 21% = 31% .31 .21 .17 38% - 21% = 17% P(TV Frigc) = 31% Chp 15 p.363 #5 P(TV XOR Refrig) = 17% + 31% = 48%
A check of dorm rooms on a large college campus revealed that 38% had refrigerators, 52% had TVs, and 21% had both a TV and a refrigerator. Whats the probability that a randomly selected dorm room has : b) A TV or a refrigerator, but not both? P(TV XOR Refrig) = 17% + 31% = 48% XOR: Exclusive OR used in Electrical Engineering. Not needed for AP Stats Chp 15 p.363 #5 P(NOT(TV U Refrig)) = = 1 P(TV U Refrig)
A check of dorm rooms on a large college campus revealed that 38% had refrigerators, 52% had TVs, and 21% had both a TV and a refrigerator. Whats the probability that a randomly selected dorm room has : c) Neither a TV nor a refrigerator? P(NOT(TV U Refrig)) = = 1 P(TV U Refrig) = 1 ( ) = = .31 Chp 15 p.363 #11 The probabilities that an adult American man has high blood pressure and /or cholesterol are shown in the table. Blood Pressure High OK 0.11 0.21 0.16 0.52 Total Cholesterol 0.32 0.68 Total 0.27 0.73 1.00 Chp 15 p.363 #11 The probabilities that an adult American man has high blood pressure and /or cholesterol are shown in the table. Blood Pressure High OK 0.11 0.21 0.16 0.52 a) Whats the probability that a man has both conditions? Total Cholesterol 0.32 0.68 Total 0.27 0.73 1.00 P(High BP High Chol) = 11% Chp 15 p.363 #11 P(High BP) = 27% High OK 0.11 0.21 0.16 0.52
The probabilities that an adult American man has high blood pressure and /or cholesterol are shown in the table. Blood Pressure High OK 0.11 0.21 0.16 0.52 b) Whats the probability that he has high blood pressure? Total Cholesterol 0.32 0.68 Total 0.27 0.73 1.00 P(High BP) = 27% Chp 15 p.363 #11 The probabilities that an adult American man has high blood pressure and /or cholesterol are shown in the table. Blood Pressure High OK 0.11 0.21 0.16 0.52 c) Whats the probability that a man with high blood pressure has high cholesterol? Total Cholesterol 0.32 0.68 Total 0.27 0.73 1.00 = . . = 40.7% Chp 15 p.363 #11 High OK 0.11 0.21 0.16
The probabilities that an adult American man has high blood pressure and /or cholesterol are shown in the table. Blood Pressure High OK 0.11 0.21 0.16 0.52 c) Whats the probability that a man with high blood pressure has high cholesterol? Total Cholesterol 0.32 0.68 Total 0.27 0.73 1.00 P(High Chol | High BP) = = . .=40.7% Chp 15 p.363 #11 The probabilities that an adult American man has high blood pressure and /or cholesterol are shown in the table. Blood Pressure High OK 0.11 0.21 0.16 0.52 d) Whats the probability that a man has high blood pressure if its known that he has high cholesterol? Cholesterol 0.32 0.68 Total 0.27 0.73 1.00 = . . = 34.37% Chp 15 p.363 #11 High OK 0.11 0.21 0.16
The probabilities that an adult American man has high blood pressure and /or cholesterol are shown in the table. Blood Pressure High OK 0.11 0.21 0.16 0.52 d) Whats the probability that a man has high blood pressure if its known that he has high cholesterol? Cholesterol 0.32 0.68 Total 0.27 0.73 1.00 P(High BP | High Chol) = = . .= % = . . = 34.37% Chp 15 p.363 #11 High OK 0.11 0.21 0.16
The probabilities that an adult American man has high blood pressure and /or cholesterol are shown in the table. Blood Pressure High OK 0.11 0.21 0.16 0.52 d) Whats the probability that a man has high blood pressure if its known that he has high cholesterol? Cholesterol 0.32 0.68 Total 0.27 0.73 1.00 P(High BP | High Chol) = = . .= % Chapter 15 Assignment pp #8&14, 22 pp #28, 36&38, 43 6. Commercial airplanes have an excellent safety record
6. Commercial airplanes have an excellent safety record. Nevertheless, there are crashes occasionally, with the loss of many lives. In the weeks following a crash, airlines often report a drop in the number of passengers, probably because people are afraid to risk flying. a) A travel agent suggests that, since the law of averages makes it highly unlikely to have two plane crashes within a few weeks of each other, flying soon after a crash is the safest time. What do you think? 6. Commercial airplanes have an excellent safety record
6. Commercial airplanes have an excellent safety record. Nevertheless, there are crashes occasionally, with the loss of many lives. In the weeks following a crash, airlines often report a drop in the number of passengers, probably because people are afraid to risk flying. b) If the airline industry proudly announces that it has set a new record for the longest period of safe flights, would you be reluctant to fly? Are the airlines due to have a crash? 8. On January 20, 2000, the International Gaming Technology company issued a press release:
(LAS VEGAS, Nev.)Cynthia Jay was smiling ear to ear as she walked into the news conference at The Desert Inn Resort in Las Vegas today, and well she should. Last night, the 37-year-old cocktail waitress won the worlds largest slot jackpot$34,959,458on a Megabucks machine. She said she had played $27 in the machine when the jackpot hit. Nevada Megabucks has produced 49 major winners in its 14-year history. The top jackpot builds from a base amount of $7 million and can be won with a 3-coin ($3) bet. a) How can the Desert Inn afford to give away millions of dollars on a $3 bet? 8. On January 20, 2000, the International Gaming Technology company issued a press release:
(LAS VEGAS, Nev.)Cynthia Jay was smiling ear to ear as she walked into the news conference at The Desert Inn Resort in Las Vegas today, and well she should. Last night, the 37-year-old cocktail waitress won the worlds largest slot jackpot$34,959,458on a Megabucks machine. She said she had played $27 in the machine when the jackpot hit. Nevada Megabucks has produced 49 major winners in its 14-year history. The top jackpot builds from a base amount of $7 million and can be won with a 3-coin ($3) bet. b) Why did the company issue a press release? Wouldnt most businesses want to keep such a huge loss quiet? 12&14. In a large Introductory Statistics lecture hall, the professor reports that 55% of the students enrolled have never taken a Calculus course, 32% have taken only one semester of Calculus, and the rest have taken two or more semesters of Calculus. The professor randomly assigns students to groups of three to work on a project for the course. i. What is the probability that the first groupmate you meet has studied a) two or more semesters of Calculus? 12&14. In a large Introductory Statistics lecture hall, the professor reports that 55% of the students enrolled have never taken a Calculus course, 32% have taken only one semester of Calculus, and the rest have taken two or more semesters of Calculus. The professor randomly assigns students to groups of three to work on a project for the course. i. What is the probability that the first groupmate you meet has studied a) two or more semesters of Calculus? 1 Sem 2+ Sem = 87 13% 32% 100 87 = 13% 55% 12&14. In a large Introductory Statistics lecture hall, the professor reports that 55% of the students enrolled have never taken a Calculus course, 32% have taken only one semester of Calculus, and the rest have taken two or more semesters of Calculus. The professor randomly assigns students to groups of three to work on a project for the course. i. What is the probability that the first groupmate you meet has studied b) some Calculus? 12&14. In a large Introductory Statistics lecture hall, the professor reports that 55% of the students enrolled have never taken a Calculus course, 32% have taken only one semester of Calculus, and the rest have taken two or more semesters of Calculus. The professor randomly assigns students to groups of three to work on a project for the course. i. What is the probability that the first groupmate you meet has studied b) some Calculus? = 68% 1 Sem 2+ Sem 13% 32% 55% 12&14. In a large Introductory Statistics lecture hall, the professor reports that 55% of the students enrolled have never taken a Calculus course, 32% have taken only one semester of Calculus, and the rest have taken two or more semesters of Calculus. The professor randomly assigns students to groups of three to work on a project for the course. i. What is the probability that the first groupmate you meet has studied c) no more than one semester of Calculus? 12&14. In a large Introductory Statistics lecture hall, the professor reports that 55% of the students enrolled have never taken a Calculus course, 32% have taken only one semester of Calculus, and the rest have taken two or more semesters of Calculus. The professor randomly assigns students to groups of three to work on a project for the course. i. What is the probability that the first groupmate you meet has studied c) no more than one semester of Calculus? = 87% 1 Sem 2+ Sem 13% 32% 55% 12&14. In a large Introductory Statistics lecture hall, the professor reports that 55% of the students enrolled have never taken a Calculus course, 32% have taken only one semester of Calculus, and the rest have taken two or more semesters of Calculus. The professor randomly assigns students to groups of three to work on a project for the course. ii. What is the probability that of your other two groupmates, a) neither has studied Calculus? 12&14. In a large Introductory Statistics lecture hall, the professor reports that 55% of the students enrolled have never taken a Calculus course, 32% have taken only one semester of Calculus, and the rest have taken two or more semesters of Calculus. The professor randomly assigns students to groups of three to work on a project for the course. ii. What is the probability that of your other two groupmates, b) both have studied at least one semester of Calculus? 12&14. In a large Introductory Statistics lecture hall, the professor reports that 55% of the students enrolled have never taken a Calculus course, 32% have taken only one semester of Calculus, and the rest have taken two or more semesters of Calculus. The professor randomly assigns students to groups of three to work on a project for the course. ii. What is the probability that of your other two groupmates, c) at least one has had more than one semester of Calculus? 18&20. A Gallup Poll in June 2004 asked 1005 U. S
18&20. A Gallup Poll in June 2004 asked 1005 U.S. adults how likely they were to read Bill Clintons autobiography My Life. Heres how they responded: Response Number Will definitely read it 90 Will probably read it 211 Will probably not read it 322 Will definitely not read it 382 i. If we select a person at random from this sample of 1005 adults, a) what is the probability that the person responded Will definitely not read it? 18&20. A Gallup Poll in June 2004 asked 1005 U. S
18&20. A Gallup Poll in June 2004 asked 1005 U.S. adults how likely they were to read Bill Clintons autobiography My Life. Heres how they responded: Response Number Will definitely read it 90 Will probably read it 211 Will probably not read it 322 Will definitely not read it 382 301 i. If we select a person at random from this sample of 1005 adults, b) what is the probability that the person will probably or definitely read it? 18&20. A Gallup Poll in June 2004 asked 1005 U. S
18&20. A Gallup Poll in June 2004 asked 1005 U.S. adults how likely they were to read Bill Clintons autobiography My Life. Heres how they responded: Response Number Will definitely read it 90 Will probably read it 211 Will probably not read it 322 Will definitely not read it 382 301 ii. Lets call someone who responded that they would definitely or probably read it a likely reader and the other two categories, unlikely readers. If we select two people at random from this sample, a) what is the probability that both are likely readers? 18&20. A Gallup Poll in June 2004 asked 1005 U. S
18&20. A Gallup Poll in June 2004 asked 1005 U.S. adults how likely they were to read Bill Clintons autobiography My Life. Heres how they responded: Response Number Will definitely read it 90 Will probably read it 211 Will probably not read it 322 Will definitely not read it 382 301 704 ii. Lets call someone who responded that they would definitely or probably read it a likely reader and the other two categories, unlikely readers. If we select two people at random from this sample, b) what is the probability that neither is a likely reader? 18&20. A Gallup Poll in June 2004 asked 1005 U. S
18&20. A Gallup Poll in June 2004 asked 1005 U.S. adults how likely they were to read Bill Clintons autobiography My Life. Heres how they responded: Response Number Will definitely read it 90 Will probably read it 211 Will probably not read it 322 Will definitely not read it 382 ii. Lets call someone who responded that they would definitely or probably read it a likely reader and the other two categories, unlikely readers. If we select two people at random from this sample, c) what is the probability that one is a likely reader and one isnt? Likely Reader Not Likely Reader (301/1005)*(301/1005) (704/1005)*(301/1005) (301/1005)*(704/1005) (704/1005)*(704/10050 18&20. A Gallup Poll in June 2004 asked 1005 U. S
18&20. A Gallup Poll in June 2004 asked 1005 U.S. adults how likely they were to read Bill Clintons autobiography My Life. Heres how they responded: Response Number Will definitely read it 90 Will probably read it 211 Will probably not read it 322 Will definitely not read it 382 ii. Lets call someone who responded that they would definitely or probably read it a likely reader and the other two categories, unlikely readers. If we select two people at random from this sample, d) What assumption did you make in computing the probabilities? 18&20. A Gallup Poll in June 2004 asked 1005 U. S
18&20. A Gallup Poll in June 2004 asked 1005 U.S. adults how likely they were to read Bill Clintons autobiography My Life. Heres how they responded: Response Number Will definitely read it 90 Will probably read it 211 Will probably not read it 322 Will definitely not read it 382 ii. Lets call someone who responded that they would definitely or probably read it a likely reader and the other two categories, unlikely readers. If we select two people at random from this sample, e) Explain why you think that assumption is reasonable. 18&20. A Gallup Poll in June 2004 asked 1005 U. S
18&20. A Gallup Poll in June 2004 asked 1005 U.S. adults how likely they were to read Bill Clintons autobiography My Life. Heres how they responded: Response Number Will definitely read it 90 Will probably read it 211 Will probably not read it 322 Will definitely not read it 382 ii. Lets call someone who responded that they would definitely or probably read it a likely reader and the other two categories, unlikely readers. If we select two people at random from this sample, e) Explain why you think that assumption is reasonable. 24. The American Red Cross says that aout 45% of the U. S
24. The American Red Cross says that aout 45% of the U.S. population has Type O blood, 40% Type A, 11% Type B, and the rest Type AB. Someone volunteers to give blood. What is the probability that this donor i. has Type AB blood? 24. The American Red Cross says that aout 45% of the U. S
24. The American Red Cross says that aout 45% of the U.S. population has Type O blood, 40% Type A, 11% Type B, and the rest Type AB. Someone volunteers to give blood. What is the probability that this donor ii. has Type A or Type B? 24. The American Red Cross says that aout 45% of the U. S
24. The American Red Cross says that aout 45% of the U.S. population has Type O blood, 40% Type A, 11% Type B, and the rest Type AB. Someone volunteers to give blood. What is the probability that this donor iii. is not Type O? 24. The American Red Cross says that aout 45% of the U. S
24. The American Red Cross says that aout 45% of the U.S. population has Type O blood, 40% Type A, 11% Type B, and the rest Type AB. b) Among four potential donors what is the probability that i. all are Type O? 24. The American Red Cross says that aout 45% of the U. S
24. The American Red Cross says that aout 45% of the U.S. population has Type O blood, 40% Type A, 11% Type B, and the rest Type AB. b) Among four potential donors what is the probability that ii. no one is Type AB? 24. The American Red Cross says that aout 45% of the U. S
24. The American Red Cross says that aout 45% of the U.S. population has Type O blood, 40% Type A, 11% Type B, and the rest Type AB. b) Among four potential donors what is the probability that iii. they are not all Type A? 24. The American Red Cross says that aout 45% of the U. S
24. The American Red Cross says that aout 45% of the U.S. population has Type O blood, 40% Type A, 11% Type B, and the rest Type AB. b) Among four potential donors what is the probability that iv. least one person is Type B? 26. Some of your answers above depended on the assumption that the outcomes described were disjoint; that is, they could not both happen at the same time. Other answers depended on the assumption that the events were independent; that is, the occurrence of one of them doesnt affect the probability of the other. Do you understand the difference between disjoint and independent? a) If you examine one person, are the events that the person is Type A and that the person is Type B disjoint or independent or neither. 26. Some of your answers above depended on the assumption that the outcomes described were disjoint; that is, they could not both happen at the same time. Other answers depended on the assumption that the events were independent; that is, the occurrence of one of them doesnt affect the probability of the other. Do you understand the difference between disjoint and independent? b) If you examine two people, are the events that the first is Type A and the second Type B disjoint or independent or neither? 26. Some of your answers above depended on the assumption that the outcomes described were disjoint; that is, they could not both happen at the same time. Other answers depended on the assumption that the events were independent; that is, the occurrence of one of them doesnt affect the probability of the other. Do you understand the difference between disjoint and independent? c) Can disjoint events ever be independent? Explain. 30. To get to work, a commuter must cross train tracks
30. To get to work, a commuter must cross train tracks. The time the train arrives varies slight from day to day, but the commuter estimates hell get stopped on about 15% of work days. During a certain 5-day work week, what is the probability that he a) gets stopped on Monday and again on Tuesday? 30. To get to work, a commuter must cross train tracks
30. To get to work, a commuter must cross train tracks. The time the train arrives varies slight from day to day, but the commuter estimates hell get stopped on about 15% of work days. During a certain 5-day work week, what is the probability that he b) gets stopped for the first time on Thursday? 30. To get to work, a commuter must cross train tracks
30. To get to work, a commuter must cross train tracks. The time the train arrives varies slight from day to day, but the commuter estimates hell get stopped on about 15% of work days. During a certain 5-day work week, what is the probability that he c) get stopped every day? 30. To get to work, a commuter must cross train tracks
30. To get to work, a commuter must cross train tracks. The time the train arrives varies slight from day to day, but the commuter estimates hell get stopped on about 15% of work days. During a certain 5-day work week, what is the probability that he d) gets stopped at least once during the week? 32. Census reports for a city indicate that 62% of residents classify themselves as Christian, 12% as Jewish, and 16% as members of other religions (Muslims, Buddhists, etc.). The remaining residents classify themselves as non-religious. A polling organization seeking information about public opinions wants to be sure to talk with people holding a variety of religious views, and makes random phone calls. Among the first four people they call, what is the probability they reach a) all Christians? 32. Census reports for a city indicate that 62% of residents classify themselves as Christian, 12% as Jewish, and 16% as members of other religions (Muslims, Buddhists, etc.). The remaining residents classify themselves as non-religious. A polling organization seeking information about public opinions wants to be sure to talk with people holding a variety of religious views, and makes random phone calls. Among the first four people they call, what is the probability they reach b) no Jews? 32. Census reports for a city indicate that 62% of residents classify themselves as Christian, 12% as Jewish, and 16% as members of other religions (Muslims, Buddhists, etc.). The remaining residents classify themselves as non-religious. A polling organization seeking information about public opinions wants to be sure to talk with people holding a variety of religious views, and makes random phone calls. Among the first four people they call, what is the probability they reach c) at least one person who is non-religious? 36. You shuffle a deck of cards, and then start turning them over one at a time. The first one is red. So is the second. And the third. In fact, you are surprised to get 10 red cards in a row. You start thinking, The next one is due to be black! a) Are you correct in thinking, that theres a higher probability that the next card will be black than red? Explain. 36. You shuffle a deck of cards, and then start turning them over one at a time. The first one is red. So is the second. And the third. In fact, you are surprised to get 10 red cards in a row. You start thinking, The next one is due to be black! b) Is this an example of the Law of Large Numbers? Explain. 36. You shuffle a deck of cards, and then start turning them over one at a time. The first one is red. So is the second. And the third. In fact, you are surprised to get 10 red cards in a row. You start thinking, The next one is due to be black! b) Is this an example of the Law of Large Numbers? Explain. Extra Practice Sixty seniors at a local high school are taking AP Statistics. Forty seniors at the same high school are taking AP Economics. If ten seniors at the high school take both classes and there are 500 seniors, answer the following questions: A. P(Stat) = Extra Practice Sixty seniors at a local high school are taking AP Statistics. Forty seniors at the same high school are taking AP Economics. If ten seniors at the high school take both classes and there are 500 seniors, answer the following questions: A. P(Stat) = = = 12% Extra Practice Sixty seniors at a local high school are taking AP Statistics. Forty seniors at the same high school are taking AP Economics. If ten seniors at the high school take both classes and there are 500 seniors, answer the following questions: B. P(Econ) = Extra Practice Sixty seniors at a local high school are taking AP Statistics. Forty seniors at the same high school are taking AP Economics. If ten seniors at the high school take both classes and there are 500 seniors, answer the following questions: B. P(Econ) = = = 8% Extra Practice Sixty seniors at a local high school are taking AP Statistics. Forty seniors at the same high school are taking AP Economics. If ten seniors at the high school take both classes and there are 500 seniors, answer the following questions: C.P( Stat Econ) = Extra Practice Sixty seniors at a local high school are taking AP Statistics. Forty seniors at the same high school are taking AP Economics. If ten seniors at the high school take both classes and there are 500 seniors, answer the following questions: C.P( Stat Econ) = = = 2% Extra Practice Sixty seniors at a local high school are taking AP Statistics. Forty seniors at the same high school are taking AP Economics. If ten seniors at the high school take both classes and there are 500 seniors, answer the following questions: P(Not Stat) = E. P(Not Econ) = Extra Practice Sixty seniors at a local high school are taking AP Statistics. Forty seniors at the same high school are taking AP Economics. If ten seniors at the high school take both classes and there are 500 seniors, answer the following questions: D. P(Not Stat) =100% 12% = 88% E. P(Not Econ) = 100% 8% = 92% Not Disjoint Extra Practice
Sixty seniors at a local high school are taking AP Statistics. Forty seniors at the same high school are taking AP Economics. If ten seniors at the high school take both classes and there are 500 seniors, answer the following questions: F. P( Stat U Econ) = P(Stat) + P(Econ) P(Stat Econ) Not Disjoint Not Disjoint Extra Practice
Sixty seniors at a local high school are taking AP Statistics. Forty seniors at the same high school are taking AP Economics. If ten seniors at the high school take both classes and there are 500 seniors, answer the following questions: F. P( Stat U Econ) = P(Stat) + P(Econ) P(Stat Econ) =12% % % = 18% Not Disjoint Extra Practice Sixty seniors at a local high school are taking AP Statistics. Forty seniors at the same high school are taking AP Economics. If ten seniors at the high school take both classes and there are 500 seniors, answer the following questions: G. P(Stat | Econ) =( ) () Extra Practice Sixty seniors at a local high school are taking AP Statistics. Forty seniors at the same high school are taking AP Economics. If ten seniors at the high school take both classes and there are 500 seniors, answer the following questions: G. P(Stat | Econ) =( ) () = =1 4 =.25 = 25% Extra Practice Sixty seniors at a local high school are taking AP Statistics. Forty seniors at the same high school are taking AP Economics. If ten seniors at the high school take both classes and there are 500 seniors, answer the following questions: H P(Econ | Stat) =( ) () Extra Practice Sixty seniors at a local high school are taking AP Statistics. Forty seniors at the same high school are taking AP Economics. If ten seniors at the high school take both classes and there are 500 seniors, answer the following questions: H P(Econ | Stat) =( ) () = =1 6 = = 16.67% Note: the P(A|B) is not always equal to the P(B|A).