Conditional Probability and Independence Example: 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 (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). Female Male Total Index finger 78 45 123 Ring finger 82 152 234 Same length 52 43 95 Total 212 240 452 Are the events “female” and “has a longer ring finger” independent? Justify your answer. Independence: A Special Multiplication Rule Multiplication rule for independent events: o If A and B are independent events, then the probability that A and B both occur is Example: 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 (bit.ly/perfectGame). So, to throw a perfect game, a pitcher needs to have nine perfect innings in a row. 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? Example: Rapid HIV Testing Many people who come to clinics to be tested for HIV, the virus that causes AIDS, don’t come back to learn the test results. Clinics now use “rapid HIV tests” that give a result while the client waits. In a clinic in Malawi, for example, use of rapid tests increased the percent of clients who learned their test results from 69% to 99.7%. The trade-off for fast results is that rapid tests are less accurate than slower laboratory tests. Applied to people who have no HIV antibodies, one rapid test has probability about 0.004 of producing a false positive (that is, of falsely indicating that antibodies are present). If a clinic tests 200 randomly selected people who are free of HIV antibodies, what is the chance that at least one false positive will occur? ***Caution***: The multiplication rule P(A and B) = P(A) · P(B) holds if A and B are independent but not otherwise. The addition rule P(A or B) = P(A) + P(B) holds if A and B are mutually exclusive but not otherwise. Resist the temptation to use these simple rules when the conditions that justify them are not present. Example: 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. Independent versus Mutually Exclusive For the following examples, determine if the two events are independent (or not) and/or mutually exclusive (or not). 1. Select one card from a standard deck and define the events A: the card is red and B: the card is a club. 2. Select one card from a standard deck and define the events A: the card is red and B: the card is a 7. 3. Select one card from a standard deck and define the events A: the card is red and B: the card is a heart.