For each problem: aIn words, define the Random Variable X size 12{X} {}. bList the values hat X may take on. cGive the distribution of X. X~ Then, answer the questions specific to each individual problem. Six different colored dice are rolled. Of interest is the number of dice that show a “1.” dOn average, how many dice would you expect to show a “1”? eFind the probability that all six dice show a “1.” fIs it more likely that 3 or that 4 dice will show a “1”? Use numbers to justify your answer numerically. a X size 12{X} {} = the number of dice that show a 1 b0,1,2,3,4,5,6 c X~B (6,16 ) d 1 e 0.00002 f 3 dice According to a 2003 publication by Waits and Lewis (source: http://nces.ed.gov/pubs2003/2003017.pdf), by the end of 2002, 92% of U.S. public two-year colleges offered distance learning courses. Suppose you randomly pick 13 U.S. public two-year colleges. We are interested in the number that offer distance learning courses. dOn average, how many schools would you expect to offer such courses? eFind the probability that at most 6 offer such courses. fIs it more likely that 0 or that 13 will offer such courses? Use numbers to justify your answer numerically and answer in a complete sentence. A school newspaper reporter decides to randomly survey 12 students to see if they will attend Tet festivities this year. Based on past years, she knows that 18% of students attend Tet festivities. We are interested in the number of students who will attend the festivities. dHow many of the 12 students do we expect to attend the festivities? eFind the probability that at most 4 students will attend. fFind the probability that more than 2 students will attend. a X size 12{X} {} = the number of students that will attend Tet. b0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 c X~B(12,0.18) d2.16 e0.9511 f0.3702 Suppose that about 85% of graduating students attend their graduation. A group of 22 graduating students is randomly chosen. dHow many are expected to attend their graduation? eFind the probability that 17 or 18 attend. fBased on numerical values, would you be surprised if all 22 attended graduation? Justify your answer numerically. At The Fencing Center, 60% of the fencers use the foil as their main weapon. We randomly survey 25 fencers at The Fencing Center. We are interested in the numbers that do not use the foil as their main weapon. dHow many are expected to not use the foil as their main weapon? eFind the probability that six do not use the foil as their main weapon. fBased on numerical values, would you be surprised if all 25 did not use foil as their main weapon? Justify your answer numerically. a X size 12{X} {} = the number of fencers that do not use foil as their main weapon b0, 1, 2, 3,... 25 c X~B(25,0.40) d10 e0.0442 fYes Approximately 8% of students at a local high school participate in after-school sports all four years of high school. A group of 60 seniors is randomly chosen. Of interest is the number that participated in after-school sports all four years of high school. dHow many seniors are expected to have participated in after-school sports all four years of high school? eBased on numerical values, would you be surprised if none of the seniors participated in after-school sports all four years of high school? Justify your answer numerically. fBased upon numerical values, is it more likely that 4 or that 5 of the seniors participated in after-school sports all four years of high school? Justify your answer numerically. The chance of having an extra fortune in a fortune cookie is about 3%. Given a bag of 144 fortune cookies, we are interested in the number of cookies with an extra fortune. Two distributions may be used to solve this problem. Use one distribution to solve the problem. dHow many cookies do we expect to have an extra fortune? eFind the probability that none of the cookies have an extra fortune. fFind the probability that more than 3 have an extra fortune. gAs n size 12{X} {} increases, what happens involving the probabilities using the two distributions? Explain in complete sentences. a X size 12{X} {} = the number of fortune cookies that have an extra fortune b0, 1, 2, 3,... 144 c X~B(25,0.40) or P(4.32) d 4.32 e 0.0124 or 0.0133 f 0.6300 or 0.6264 There are two games played for Chinese New Year and Vietnamese New Year. They are almost identical. In the Chinese version, fair dice with numbers 1, 2, 3, 4, 5, and 6 are used, along with a board with those numbers. In the Vietnamese version, fair dice with pictures of a gourd, fish, rooster, crab, crayfish, and deer are used. The board has those six objects on it, also. We will play with bets being $1. The player places a bet on a number or object. The “house” rolls three dice. If none of the dice show the number or object that was bet, the house keeps the $1 bet. If one of the dice shows the number or object bet (and the other two do not show it), the player gets back his $1 bet, plus $1 profit. If two of the dice show the number or object bet (and the third die does not show it), the player gets back his $1 bet, plus $2 profit. If all three dice show the number or object bet, the player gets back his $1 bet, plus $3 profit. Let X size 12{X} {} = number of matches and Y size 12{Y} {}= profit per game.dList the values that Y size 12{Y} {} may take on. Then, construct one PDF table that includes both X size 12{X} {} & Y size 12{Y} {} and their probabilities. eCalculate the average expected matches over the long run of playing this game for the player. fCalculate the average expected earnings over the long run of playing this game for the player. gDetermine who has the advantage, the player or the house. According to the South Carolina Department of Mental Health web site, for every 200 U.S. women, the average number who suffer from anorexia is one (http://www.state.sc.us/dmh/anorexia/statistics.htm). Out of a randomly chosen group of 600 U.S. women: dHow many are expected to suffer from anorexia? eFind the probability that no one suffers from anorexia. fFind the probability that more than four suffer from anorexia. a X size 12{X} {} = the number of women that suffer from anorexia b0, 1, 2, 3,... 600 (can leave off 600) c X~P(3) d3 e0.0498 f0.1847 The average number of children of middle-aged Japanese couples is 2.09 (Source: The Yomiuri Shimbun, June 28, 2006). Suppose that one middle-aged Japanese couple is randomly chosen. dFind the probability that they have no children. eFind the probability that they have fewer children than the Japanese average. fFind the probability that they have more children than the Japanese average . The average number of children per Spanish couples was 1.34 in 2005. Suppose that one Spanish couple is randomly chosen. (Source: http://www.typicallyspanish.com/news/publish/article_4897.shtml, June 16, 2006). dFind the probability that they have no children. eFind the probability that they have fewer children than the Spanish average. fFind the probability that they have more children than the Spanish average . a X size 12{X} {} = the number of children for a Spanish couple b0, 1, 2, 3,... c X~P(1.34) d0.2618 e0.6217 f0.3873 Fertile (female) cats produce an average of 3 litters per year. (Source: The Humane Society of the United States). Suppose that one fertile, female cat is randomly chosen. In one year, find the probability she produces: dNo litters. eAt least 2 litters. fExactly 3 litters. A consumer looking to buy a used red Miata car will call dealerships until she finds a dealership that carries the car. She estimates the probability that any independent dealership will have the car will be 28%. We are interested in the number of dealerships she must call. dOn average, how many dealerships would we expect her to have to call until she finds one that has the car? eFind the probability that she must call at most 4 dealerships. fFind the probability that she must call 3 or 4 dealerships. a X size 12{X} {} = the number of dealers she calls until she finds one with a used red Miata b0, 1, 2, 3,... c X~G(0.28) d3.57 e0.7313 f0.2497 Suppose that the probability that an adult in America will watch the Super Bowl is 40%. Each person is considered independent. We are interested in the number of adults in America we must survey until we find one who will watch the Super Bowl. dHow many adults in America do you expect to survey until you find one who will watch the Super Bowl? eFind the probability that you must ask 7 people. fFind the probability that you must ask 3 or 4 people. A group of Martial Arts students is planning on participating in an upcoming demonstration. 6 are students of Tae Kwon Do; 7 are students of Shotokan Karate. Suppose that 8 students are randomly picked to be in the first demonstration. We are interested in the number of Shotokan Karate students in that first demonstration. dHow many Shotokan Karate students do we expect to be in that first demonstration? eFind the probability that 4 students of Shotokan Karate are picked. fFind the probability that no more than 6 students of Shotokan Karate are picked. d4.31 e0.4079 f0.9953 The chance of a IRS audit for a tax return with over $25,000 in income is about 2% per year. We are interested in the expected number of audits a person with that income has in a 20 year period. Assume each year is independent. dHow many audits are expected in a 20 year period? eFind the probability that a person is not audited at all. fFind the probability that a person is audited more than twice. Refer to the previous problem. Suppose that 100 people with tax returns over $25,000 are randomly picked. We are interested in the number of people audited in 1 year. One way to solve this problem is by using the Binomial Distribution. Since n is large and p is small, another discrete distribution could be used to solve the following problems. Solve the following questions (d-f) using that distribution. dHow many are expected to be audited? eFind the probability that no one was audited. fFind the probability that more than 2 were audited. d2 e0.1353 f0.3233 Suppose that a technology task force is being formed to study technology awareness among instructors. Assume that 10 people will be randomly chosen to be on the committee from a group of 28 volunteers, 20 who are technically proficient and 8 who are not. We are interested in the number on the committee who are not technically proficient. dHow many instructors do you expect on the committee who are not technically proficient? eFind the probability that at least 5 on the committee are not technically proficient. fFind the probability that at most 3 on the committee are not technically proficient. Refer back to Exercise 4.15.12. Solve this problem again, using a different, though still acceptable, distribution. a X size 12{X} {} = the number of seniors that participated in after-school sports all 4 years of high school b0, 1, 2, 3,... 60 c X ~ P ( 4 . 8 ) size 12{X "~" P $ 4 "." 8 $ } {} d4.8 eYes f4 Suppose that 9 Massachusetts athletes are scheduled to appear at a charity benefit. The 9 are randomly chosen from 8 volunteers from the Boston Celtics and 4 volunteers from the New England Patriots. We are interested in the number of Patriots picked. dIs it more likely that there will be 2 Patriots or 3 Patriots picked? eWhat is the probability that all of the volunteers will be from the Celtics fIs it more likely that more of the volunteers will be from the Patriots or from the Celtics? How do you know? On average, Pierre, an amateur chef, drops 3 pieces of egg shell into every 2 batters of cake he makes. Suppose that you buy one of his cakes. dOn average, how many pieces of egg shell do you expect to be in the cake? eWhat is the probability that there will not be any pieces of egg shell in the cake? fLet’s say that you buy one of Pierre’s cakes each week for 6 weeks. What is the probability that there will not be any egg shell in any of the cakes? gBased upon the average given for Pierre, is it possible for there to be 7 pieces of shell in the cake? Why? a X size 12{X} {} = the number of shell pieces in one cake b0, 1, 2, 3,... c X ~ P ( 1 . 5 ) size 12{X "~" P $ 1 "." 5 $ } {} d1.5 e0.2231 f0.0001 gYes It has been estimated that only about 30% of California residents have adequate earthquake supplies. Suppose we are interested in the number of California residents we must survey until we find a resident who does not have adequate earthquake supplies. dWhat is the probability that we must survey just 1 or 2 residents until we find a California resident who does not have adequate earthquake supplies? eWhat is the probability that we must survey at least 3 California residents until we find a California resident who does not have adequate earthquake supplies? fHow many California residents do you expect to need to survey until you find a California resident who does not have adequate earthquake supplies? gHow many California residents do you expect to need to survey until you find a California resident who does have adequate earthquake supplies? Refer to the above problem. Suppose you randomly survey 11 California residents. We are interested in the number who have adequate earthquake supplies. d What is the probability that at least 8 have adequate earthquake supplies? eIs it more likely that none or that all of the residents surveyed will have adequate earthquake supplies? Why? fHow many residents do you expect will have adequate earthquake supplies? d0.0043 enone f3.3 The next 3 questions refer to the following: In one of its Spring catalogs, L.L. Bean® advertised footwear on 29 of its 192 catalog pages. Suppose we randomly survey 20 pages. We are interested in the number of pages that advertise footwear. Each page may be picked at most once. dHow many pages do you expect to advertise footwear on them? eIs it probable that all 20 will advertise footwear on them? Why or why not? fWhat is the probability that less than 10 will advertise footwear on them? Suppose we randomly survey 20 pages. We are interested in the number of pages that advertise footwear. This time, each page may be picked more than once. dHow many pages do you expect to advertise footwear on them? eIs it probable that all 20 will advertise footwear on them? Why or why not? f What is the probability that less than 10 will advertise footwear on them? gSuppose that a page may be picked more than once. We are interested in the number of pages that we must randomly survey until we find one that has footwear advertised on it. Define the random variable X and give its distribution. hDo you expect to survey more than 10 pages in order to find one that advertises footwear on it? Why? iWhat is the probability that you only need to survey at most 3 pages in order to find one that advertises footwear on it? jHow many pages do you expect to need to survey in order to find one that advertises footwear? d3.02 eNo f0.9997 h0.2291 i0.3881 j6.6207 pages