Notes 5.3 Conditional Probability and Independence
Who owns a home? Phone usage 1. If we know that a person owns a home, what is the probability that the person is a high school graduate? There are a total of 340 people in the sample who own a home. Because there are 221 high school graduates among the 340 homeowners, the desired probability is 2. If we know that a person is a high school graduate, what is the probability that the person owns a home? There are a total of 310 people who are high school graduates. Because there are 221 homeowners among the 310 high school graduates, the desired probability is P( Problem: What is the probability that a randomly selected household with a landline also has a cell phone? P(o Playing in the NCAA About 55% of high school students participate in a school athletic team at some level, and about 5% of these athletes go on to play on a college team in the NCAA (http://www.washingtonpost.com/wp-dyn/content/article/2009/09/23/ AR2009092301947.html, http://www.collegesportsscholarships.com/percentage-high-schoolathletes-ncaa-college.htm). Problem: What percent of high school students play a sport in high school and go on to play a sport in the NCAA?
Late for school Shannon hits the snooze bar on her alarm clock on 60% of school days. If she doesn t hit the snooze bar, there is a 0.90 probability that she makes it to class on time. However, if she hits the snooze bar, there is only a 0.70 probability that she makes it to class on time. On a randomly chosen day, what is the probability that Shannon is late for class? Suppose that Shannon is late for school. What is the probability that she hit the snooze bar that morning? Media usage and good grades The Kaiser Family Foundation recently released a study about the influence of media in the lives of young people aged 8 18 (www.kff.org/entmedia/mh012010pkg.cfm). In the study, 17% of the youth were classified as light media users, 62% were classified as moderate media users, and 21% were classified as heavy media users. Of the light users who responded, 74% described their grades as good (A s and B s), while only 68% of the moderate users and 52% of the heavy users described their grades as good. Suppose that we selected one young person at random. Problem: (a) Draw a tree diagram to represent this situation. (b) Find the probability that this person describes his or her grades as good. (c) Given that this person describes his or her grades as good, what is the probability that he or she is a heavy user of media? False positives and drug testing Many employers require prospective employees to take a drug test. A positive result on this test indicates that the prospective employee uses illegal drugs. However, not all people who test positive actually use drugs. Suppose that 4% of prospective employees use drugs, the false positive rate is 5%, and the false negative rate is 10%. (http://www.cbsnews.com/stories/2010/06/01/health/webmd/ main6537635.shtml). Problem: A randomly selected prospective employee tests positive for drugs. What is the probability that he actually took drugs?
Ponder this: 32 gogurts come in a box ( 16 berry, 16 strawberry). Son #1 states he always grabs berry Son #2 hates berry, only eats strawberry Son #3 has no clue what he wants/eats Question: P(son #1 grabs 3 berrys for my 3 boys) Independent, Mutually Exclusive, Neither?? Draw one card 1. P(red club) Things to consider: P(red)= Who owns a home? The events of interest in this scenario were G: is a high school graduate and H: owns a home. We already learned that P(H) = 340/500 = 68% and that P(H G) = 221/310 = 71.3%. That is, we know that a randomly selected member of the sample has a 68% probability of owning a home. However, if we know that the randomly selected member is a high school graduate, the probability of owning a home increases to 71.3%. Because knowing the outcome of one event changes the probability of the other event, these two events are not independent. P(red club)= 2. P(red 7) Things to consider: P(red)= P(red 7)= 3. P(red heart) Things to consider: P(red)= P(red heart)=
Finger length Is there a relationship between gender and relative finger length? To find out, we used the random sampler at the United States CensusAtSchool Web site (www.amstat.org/censusatschool) to randomly select 452 U.S. high school students who completed a survey. The two-way table shows the gender of each student and which finger was longer on their left hand (index finger or ring finger). Problem: Are the events female and has a longer ring finger independent? Justify your answer. Perfect games In baseball, a perfect game is when a pitcher doesn t allow any hitters to reach base in all nine innings. Historically, pitchers throw a perfect inning an inning where no hitters reach base about 40% of the time (http://www.baseballprospectus.com/article.php? articleid=11110). So, to throw a perfect game, a pitcher needs to have nine perfect innings in a row. Problem: What is the probability that a pitcher throws nine perfect innings in a row, assuming the pitcher s performance in an inning is independent of his performance in other innings? Weather conditions Hacienda Heights and La Puente are two neighboring suburbs in the Los Angeles area. According to the local newspaper, there is a 50% chance of rain tomorrow in Hacienda Heights and a 50% chance of rain in La Puente. Does this mean that there is a (0.5)(0.5) = 0.25 probability that it will rain in both cities tomorrow? No. It is not appropriate to multiply the two probabilities, because the events aren t independent. If it is raining in one of these locations, there is a very high probability that it is raining in the other location. However, suppose that there was also a 50% chance of rain in New York tomorrow. To find the probability that it will rain in Hacienda Heights and in New York, it would be appropriate to multiply the probabilities, because it is reasonable to believe that knowledge of rain in Hacienda Heights won t help us predict rain in New York.
HW: 57-60, 63, 66, 67, 72, 74, 78, 80 HW: 82, 83, 86, 89, 92, 93, 95, 97-99