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Chapter 5 Section 3 day 3 2014f.notebook November 21, 2014 Honors Statistics Friday November 21, 2014 1 Chapter 5 Section 3 day 3 2014f.notebook November 21, 2014 Daily Agenda 1. Welcome to class 2. Please find folder and take your seat. 3. Review Homework C5#7 4. Chapter 5 Section 3 Tree diagrams 5. Homework announcement 6. Collect folders 2 Chapter 5 Section 3 day 3 2014f.notebook November 21, 2014 3 Chapter 5 Section 3 day 3 2014f.notebook November 21, 2014 4 Chapter 5 Section 3 day 3 2014f.notebook November 21, 2014 5 Chapter 5 Section 3 day 3 2014f.notebook November 21, 2014 6 Chapter 5 Section 3 day 3 2014f.notebook November 21, 2014 7 Chapter 5 Section 3 day 3 2014f.notebook November 21, 2014 OTL C5#7 homework assignment How do you want it - the crystal mumbo-jumbo or statistical probability? USE PROPER NOTATION !!!! P( ...) = page 333 ­ 334: 63 to 73 (11 problems) Contest winners only do the ODD #'s 8 Chapter 5 Section 3 day 3 2014f.notebook November 21, 2014 Exercise C5#63: Get Rich page 333 Get rich A survey of 4826 randomly selected young adults (aged 19 to 25) asked, “What do you think are the chances you will have much more than a middle­class income at age 30?” The two­way table shows the responses.16 Choose a survey respondent at random. (a) Given that the person selected is male, what’s the probability that he answered “almost certain”? 597 P(AC I M) = ­­­­­­­­­ = 0.2428 2459 (b) If the person selected said “some chance but probably not,” what’s the probability that the person is female? 426 P(F I ScPn) = ­­­­­­­­­ = 0.5983 712 9 Chapter 5 Section 3 day 3 2014f.notebook November 21, 2014 Exercise C5#64: A Titanic disaster page 333 319 261 627 442 765 1207 A Titanic disaster In 1912 the luxury liner Titanic, on its first voyage across the Atlantic, struck an iceberg and sank. Some passengers got off the ship in lifeboats, but many died. The two­way table gives information about adult passengers who lived and who died, by class of travel. Suppose we choose an adult passenger at random. > (a) Given that the person selected was in first class, what’s the probability that he or she survived? 197 P(Survived I FC) = ­­­­­­­­­ = 0.6176 319 > (b) If the person selected survived, what’s the probability that he or she was a third­class passenger? 151 P( third class I Survived) = ­­­­­­­­­ = 0.3416 442 10 Chapter 5 Section 3 day 3 2014f.notebook November 21, 2014 Exercise C5#65: Senators page 333 60 40 83 17 100 Sampling senators The two­way table describes the members of the U.S. Senate in a recent year. Suppose we select a senator at random. Consider events D: is a democrat, and F: is female. (a) Find P(D | F). Explain what this value means. 13 P(D I F) = ­­­­­­­­­ = 0.765 17 What is the probability that you are a democrat given that you are a female senator? 76.5% (b) Find P(F | D). Explain what this value means. 13 P(F I D) = ­­­­­­­­­ = 0.217 What is the probability that you are a female given that you are a democratic senator? 21.7% 60 ARE THESE EVENT INDEPENDENT? ARE GENDER AND PARTY Independent events for the U.S. Senators? 11 Chapter 5 Section 3 day 3 2014f.notebook November 21, 2014 Exercise C5#66: Who eats breakfast ? page 333 Who eats breakfast? The following two­way table describes the 595 students who responded to a school survey about eating breakfast. Suppose we select a student at random. Consider events B: eats breakfast regularly, and M: is male. (a) Find P(B | M). Explain what this value means. 190 P(B I M) = ­­­­­­­­­ = 0.5938 320 What is the probability that you are a male given that you eat breakfast regularly 59.38% (b) Find P(M | B). Explain what this value means. 190 P(M I B) = ­­­­­­­­­ = 0.6333 What is the probability that you eat breakfast regularly given that you are a male? 63.33% 300 ARE THESE EVENT INDEPENDENT? ARE GENDER AND Breakfast habits Independent events? 12 Chapter 5 Section 3 day 3 2014f.notebook November 21, 2014 Exercise C5#67: Foreign­language study page 334 Foreign­language study Choose a student in grades 9 to 12 at random and ask if he or she is studying a language other than English. Here is the distribution of results: > (a) What’s the probability that the student is studying a language other than English? P(other than English) = 1 = P(none) = 1 ­ 0.59 = 0.41 > (b) What is the conditional probability that a student is studying Spanish given that he or she is studying some language other than English? 0.26 P( S I other than English) = ­­­­­­­­ = 0.6341 0.41 13 Chapter 5 Section 3 day 3 2014f.notebook November 21, 2014 Exercise C5#68: Income tax returns page 334 Income tax returns Here is the distribution of the adjusted gross income (in thousands of dollars) reported on individual federal income tax returns in a recent year: > (a) What is the probability that a randomly chosen return shows an adjusted gross income of $50,000 or more? P(income ≥ 50000) = 0.215 + 0.100 + 0.006 = 0.321 > (b) Given that a return shows an income of at least $50,000, what is the conditional probability that the income is at least $100,000? 0.106 P( income ≥ 100,000 I income ≥ 50,000) = ­­­­­­­­­­­ = 0.3302 0.321 14 Chapter 5 Section 3 day 3 2014f.notebook November 21, 2014 Exercise C5#70: Teachers and college degrees page 334 Teachers and college degrees Select an adult at random. Define events A: person has earned a college degree, and T: person’s career is teaching. Rank the following probabilities from smallest to largest. Justify your answer. P(A) P(T) P(A | T) P(T | A) P(T)... some of the population of adults are teachers P(T I A) ... some people who have college degrees are teachers P(A) ... many people have college degrees P(A I T) ... all teachers have college degrees 15 Chapter 5 Section 3 day 3 2014f.notebook November 21, 2014 Exercise C5#69: Tall people and basketball players page 334 Tall people and basketball players Select an adult at random. Define events T: person is over 6 feet tall, and B: person is a professional basketball player. Rank the following probabilities from smallest to largest. Justify your answer. P(T) P(B) P(T | B) P(B | T) P(B) ... some people play basketball (few pro) P(B I T) ... some tall people play basketball P(T)... more of the population of adults are tall P(T I B) ... almost all pro basketball players are tall 16 Chapter 5 Section 3 day 3 2014f.notebook November 21, 2014 Exercise C5#71: Facebook versus YouTube page 334 Facebook versus YouTube A recent survey suggests that 85% of college students have posted a profile on Facebook, 73% use YouTube regularly, and 66% do both. Suppose we select a college student at random and learn that the student has a profile on Facebook. Find the probability that the student uses YouTube regularly. Show your work. FACE N F UTube 0.66 0.07 0.73 N U 0.19 0.08 0.27 0.85 0.15 1.00 0.66 P( UTube I Face) = ­­­­­­­­­­ = 0.7765 0.85 17 Chapter 5 Section 3 day 3 2014f.notebook November 21, 2014 Exercise C5#72: MAC or PC? page 334 Mac or PC? A recent census at a major university revealed that 40% of its students mainly used Macintosh computers (Macs). The rest mainly used PCs. At the time of the census, 67% of the school’s students were undergraduates. The rest were graduate students. In the census, 23% of the respondents were graduate students who said that they used PCs as their primary computers. Suppose we select a student at random from among those who were part of the census and learn that the student mainly uses a PC. Find the probability that this person is a graduate student. Show your work. MAC PC 0.30 0.37 0.67 Grad 0.10 0.23 0.33 0.40 0.60 1.00 Und 0.23 P(G I PC) = ­­­­­­­­­­ = 0.383 0.60 18 Chapter 5 Section 3 day 3 2014f.notebook November 21, 2014 Exercise C5#73: Free downloads? page 334 Free downloads? Illegal music downloading has become a big problem: 29% of Internet users download music files, and 67% of downloaders say they don’t care if the music is copyrighted.17 What percent of Internet users download music and don’t care if it’s copyrighted? Write the information given in terms of probabilities, and use the general multiplication rule. P( IDK and download) P( IDK I download) = ­­­­­­­­­­­­­­­­­ P(download) x 0.67 = ­­­­­­­­­­ 0.29 x = (0.67)(0.29) x = 0.1943 P( IDK and download) = 0.1943 or 19.4% 19 Chapter 5 Section 3 day 3 2014f.notebook November 21, 2014 Please place the following in your pocket folder: 1. your quiz and test (into folder) 2. your homework papers into binder Please put your folders on the back shelf at the end of the class or as directed by your teacher 20 Chapter 5 Section 3 day 3 2014f.notebook November 21, 2014 21 Chapter 5 Section 3 day 3 2014f.notebook November 21, 2014 22 Chapter 5 Section 3 day 3 2014f.notebook November 21, 2014 23 Chapter 5 Section 3 day 3 2014f.notebook November 21, 2014 24 Chapter 5 Section 3 day 3 2014f.notebook November 21, 2014 25 Chapter 5 Section 3 day 3 2014f.notebook November 21, 2014 26 Chapter 5 Section 3 day 3 2014f.notebook November 21, 2014 27 Chapter 5 Section 3 day 3 2014f.notebook November 21, 2014 28 Chapter 5 Section 3 day 3 2014f.notebook November 21, 2014 29 Chapter 5 Section 3 day 3 2014f.notebook November 21, 2014 30 Chapter 5 Section 3 day 3 2014f.notebook November 21, 2014 What percent of people actually have Lyme Disease, given they have tested positive? 31 Chapter 5 Section 3 day 3 2014f.notebook November 21, 2014 32 Chapter 5 Section 3 day 3 2014f.notebook November 21, 2014 33 Chapter 5 Section 3 day 3 2014f.notebook November 21, 2014 34 Chapter 5 Section 3 day 3 2014f.notebook November 21, 2014 Exercise C5#81: HIV testing page 335 HIV testing Enzyme immunoassay (EIA) tests are used to screen blood specimens for the presence of antibodies to HIV, the virus that causes AIDS. Antibodies indicate the presence of the virus. The test is quite accurate but is not always correct. Here are approximate probabilities of positive and negative EIA outcomes when the blood tested does and does not actually contain antibodies to HIV:21 Suppose that 1% of a large population carries antibodies to HIV in their blood. We choose a person from this population at random. Given that the EIA test is positive, find the probability that the person has the antibody. Show your work. 35 Chapter 5 Section 3 day 3 2014f.notebook November 21, 2014 Exercise C5#81: HIV testing page 335 HIV testing Enzyme immunoassay (EIA) tests are used to screen blood specimens for the presence of antibodies to HIV, the virus that causes AIDS. Antibodies indicate the presence of the virus. The test is quite accurate but is not always correct. Here are approximate probabilities of positive and negative EIA outcomes when the blood tested does and does not actually contain antibodies to HIV:21 Suppose that 1% of a large population carries antibodies to HIV in their blood. We choose a person from this population at random. Given that the EIA test is positive, find the probability that the person has the antibody. Show your work. 36 Chapter 5 Section 3 day 3 2014f.notebook November 21, 2014 INDEPENDENCE IS NOT DISJOINT DISJOINT SETS could be Independent IF two sets are independent their "picture" is intersecting. IF a picture of two sets is intersecting the two sets may or may not be independent. If two sets are independent (set A and set B) then... Knowing something about set A does nothing to help me decide if the "item" is in set B Potential independent situation .... GIRLS BOYS Male Staff member 37 Chapter 5 Section 3 day 3 2014f.notebook November 21, 2014 IF two sets are independent this equation must be TRUE P(A ∩ B) = P(A)P(B) .4 .2 .1 .3 does .2 = .6 x .3 NO .2 ≠ .18 so sets are not independent can you change the probabilities and make the sets independent? OR Does P(A) = P(A I B) ??? 0.2 Does 0.6 = ­­­­­­­ = 0.666 NO 0.3 .5 .3 .1 .1 Let's try again ... P(B) = P(B I A) ?? 0.3 0.4 = ­­­­­­­ = 0.375 NO 0.8 38 Chapter 5 Section 3 day 3 2014f.notebook November 21, 2014 A skips 75, 78, 91 OTL C5#8 homework assignment How do you want it - the crystal mumbo-jumbo or statistical probability? USE PROPER NOTATION !!!! P( ...) = page 334­337: 75t, 76t, 77t, 78t, 80, 89, 91, 92, 94, 96 (10 problems) t = tree diagram DO IT!! 39 Chapter 5 Section 3 day 3 2014f.notebook November 21, 2014 Exercise C5#75: Box of Chocolates page 334 Box of chocolates According to Forrest Gump, “Life is like a box of chocolates. You never know what you’re gonna get.” Suppose a candy maker offers a special “Gump box” with 20 chocolate candies that look the same. In fact, 14 of the candies have soft centers and 6 have hard centers. Choose 2 of the candies from a Gump box at random. (a) Draw a tree diagram that shows the sample space of this chance process. 13 19 soft hard 14 20 soft 20 chocolates 14 soft 6 hard centers 6 P(soft ∩ soft)= 0.4789 P(soft ∩ hard)= 0.2211 P(hard ∩ soft)= 0.2211 P(hard ∩ hard)= 0.0789 19 14 hard 6 20 19 soft hard 5 19 (b) Find the probability that one of the chocolates has a soft center and the other one doesn’t. P(one soft and one hard) = 0.2211 + 0.2211 = 0.4422 40 Chapter 5 Section 3 day 3 2014f.notebook November 21, 2014 Exercise C5#76: Inspecting switches page 334 Inspecting switches A shipment contains 10,000 switches. Of these, 1000 are bad. An inspector draws 2 switches at random, one after the other. (a) Draw a tree diagram that shows the sample space of this chance process. 8999 9999 9000 10000 10,000 switches 1000 bad good P(good ∩ good)= good bad P(good ∩ bad)= 1000 0.8099 0.0900 9999 9000 bad 1000 10000 9999 good bad 999 P(bad ∩ good)= 0.0900 P(bad ∩ bad)= 0.00999 9999 (b) Find the probability that both switches are defective. P(bad ∩ bad)= = 0.00999 41 Chapter 5 Section 3 day 3 2014f.notebook November 21, 2014 Exercise C5#77: Fill'er up! page 334 Fill ‘er up! In a recent month, 88% of automobile drivers filled their vehicles with regular gasoline, 2% purchased midgrade gas, and 10% bought premium gas.18 Of those who bought regular gas, 28% paid with a credit cardÍž of customers who bought midgrade and premium gas, 34% and 42%, respectively, paid with a credit card. Suppose we select a customer at random. (a) Draw a tree diagram to represent this situation. .28 credit no cre dit .72 20­ P(reg ∩ credit)= (0.88)(0.28) = 0.2464 P(reg ∩ no credit)=(0.88)(0.72) = 0.6336 regular .88 .34 midgrade .02 credit no cre dit .66 20­ P(mid ∩ credit)= (0.02)(0.34) =0.0068 P(mid ∩ no credit)=(0.02)(0.66) = 0.0132 premium .10 .42 credit no cre dit .58 20­ P(prem ∩ credit)= (0.10)(0.42) = 0.042 P(prem ∩ no credit)=(0.10)(0.58) = 0.058 (b) Find the probability that the customer paid with a credit card. Show your work. P(credit) = 0.2464 + 0.0068 + 0.042 = 0.2952 (c) Given that the customer paid with a credit card, find the probability that she bought premium gas. Show your work. 0.042 P(premium I credit) = ­­­­­­­ = 0.142 0.2952 42 Chapter 5 Section 3 day 3 2014f.notebook November 21, 2014 Exercise C5#78: Urban Voters page 335 Urban voters The voters in a large city are 40% white, 40% black, and 20% Hispanic. (Hispanics may be of any race in official statistics, but here we are speaking of political blocks.) A mayoral candidate anticipates attracting 30% of the white vote, 90% of the black vote, and 50% of the Hispanic vote. Suppose we select a voter at random. Can you tell what city elected this candidate for Mayor? (a) Draw a tree diagram to represent this situation. P(White ∩ vote +) = 0.12 0.30 vote for ite 40 0. h W Black 0.40 Voters vote again st 0.7 Hi sp again P(White ∩ vote ­) = 0.28 P(Black ∩ vote +) = 0.36 r 0.90 vote fo vote 0 st 0.1 0 P(Black ∩ vote +) = 0.04 an ic 0. 20 0.50 vote for vote again s t 0.50 P(Hispanic ∩ vote +) = 0.10 P(Hispanic ∩ vote ­) = 0.10 (b) Find the probability that this voter votes for the mayoral candidate. Show your work. P(votes for the mayoral candidate) = 0.12 + 0.36 + 0.10 = 0.58 (c) Given that the chosen voter plans to vote for the candidate, find the probability that the voter is black. Show your work. 0.36 P(Black voter I vote for mayoral candidate) = ­­­­­­­ = 0.62 0.58 43 Chapter 5 Section 3 day 3 2014f.notebook November 21, 2014 Exercise C5#80: Urban Voters page 335 Fundraising by telephone Tree diagrams can organize problems having more than two stages. The figure at top right shows probabilities for a charity calling potential donors by telephone.20 Each person called is either a recent donor, a past donor, or a new prospect. At the next stage, the person called either does or does not pledge to contribute, with conditional probabilities that depend on the donor class to which the person belongs. Finally, those who make a pledge either do or don’t actually make a contribution. Suppose we randomly select a person who is called by the charity. > a) What is the probability that the person contributed to the charity? Show your work. > (b) Given that the person contributed, find the probability that he or she is a recent donor. Show your work. 44 Chapter 5 Section 3 day 3 2014f.notebook November 21, 2014 Exercise C5#89: Urban Voters page 335 Bright lights? A string of Christmas lights contains 20 lights. The lights are wired in series, so that if any light fails, the whole string will go dark. Each light has probability 0.02 of failing during a 3­year period. The lights fail independently of each other. Find the probability that the string of lights will remain bright for 3 years. P(lights state lit) = 1­P(one goes out) = 1­.02 = .98 but 20 lights sooooo P(lit) = .98 20 = .668 45 Chapter 5 Section 3 day 3 2014f.notebook November 21, 2014 Exercise C5#91: Urban Voters page 335 Universal blood donors People with type O­negative blood are universal donors. That is, any patient can receive a transfusion of O­negative blood. Only 7.2% of the American population have O­negative blood. If we choose 10 Americans at random who gave blood, what is the probability that at least 1 of them is a universal donor? 46 Chapter 5 Section 3 day 3 2014f.notebook November 21, 2014 Exercise C5#92: Urban Voters page 335 Lost Internet sites Internet sites often vanish or move, so that references to them can’t be followed. In fact, 13% of Internet sites referenced in major scientific journals are lost within two years after publication.22 If we randomly select seven Internet references, from scientific journals, what is the probability that at least one of them doesn’t work two years later? 47 Chapter 5 Section 3 day 3 2014f.notebook November 21, 2014 Exercise C5#94: Urban Voters page 335 Late flights An airline reports that 85% of its flights arrive on time. To find the probability that its next four flights into LaGuardia Airport all arrive on time, can we multiply (0.85)(0.85)(0.85)(0.85)? Why or why not? 48 Chapter 5 Section 3 day 3 2014f.notebook November 21, 2014 Exercise C5#96: Urban Voters page 335 The probability of a flush A poker player holds a flush when all 5 cards in the hand belong to the same suit. We will find the probability of a flush when 5 cards are dealt. Remember that a deck contains 52 cards, 13 of each suit, and that when the deck is well shuffled, each card dealt is equally likely to be any of those that remain in the deck. (a) We will concentrate on spades. What is the probability that the first card dealt is a spade? What is the conditional probability that the second card is a spade given that the first is a spade? P(spade) = 13/52 = .25 P(spade l spade) = (12/51) = .23529 (b) Continue to count the remaining cards to find the conditional probabilities of a spade on the third, the fourth, and the fifth card given in each case that all previous cards are spades. P(spade l spade l spade) = (11/50) = .22 P(spade l spade l spade l spade) = (10/49) =.2041 P(spade l spade l spade l spade l spade) = (9/48) = .1875 (c) The probability of being dealt 5 spades is the product of the five probabilities you have found. Why? What is this probability? using the extended multiplication rule gives the probability of a flush P(spade flush) = (.25)(.2353)(.22)(.2041)(.1875) = .0004952 (d) The probability of being dealt 5 hearts or 5 diamonds or 5 clubs is the same as the probability of being dealt 5 spades. What is the probability of being dealt a flush? multiply answer c by 4 (4)(.0004952) = .001981 about 2 out of 1000 hands (in the looooong run) 49 Chapter 5 Section 3 day 3 2014f.notebook November 21, 2014 50