6.2 Independence and the Multiplication Rule

Independence and Multiplication Toss 2 coins: A = first is heads B = second is tails Are these disjoint? Venn Diagram: Does A change the outcome of B? Or vice versa?

Independence and Multiplication independent events— If independent, then:

Suppose that 40% of a population is Democrat, 40% Republican, and 20% Independent. Also, suppose our population is 30% male and 70% female. For convenience, suppose the population of interest consists of 1000 people. Then a two-way table would look like this so far:   Democrat Republican Independent TOTAL Men 300 Women 700 400 200 1000

If independent: Democrat Republican Independent TOTAL Men 300 Women   Democrat Republican Independent TOTAL Men 300 Women 700 400 200 1000

If male and democrat are mututally exclusive:   Democrat Republican Independent TOTAL Men 300 Women 700 400 200 1000

Notes about Independence

Practice Problems: pg. 427 6.19 Sudden infant death syndrome (SIDS) causes babies to die suddenly (often in their cribs), but as of yet no cause is known. Deaths from SIDS have been greatly reduced by placing babies on their backs. When more than one SIDS death occurs in a family, the parents are sometimes accused. One “expert witness” popular with prosecutors in England told juries that there is only a 1 in 72 million chance that two children in the same family could have died naturally. The rate of SIDS in a nonsmoking middle-class family is 1 in 8500. So the probability of two deaths is:

Practice Problems: pg. 427 6.19 Several women were convicted of murder on this basis, without any direct evidence that they harmed their children. As the Royal Statistical Society said, this reasoning is nonsense. It assumes that SIDS deaths in the same family are independent events. The cause of SIDS is unknown. “There may well be unknown genetic or environmental factors that predispose families to SIDS, so that a second case within the family becomes more likely.” The British government decided to review the cases of 258 parents convicted of murdering their babies.

Pg. 430 #48 A= {The person chosen is male} Age Group Female Male Total 15-17 89 61 150 18-24 5668 4697 10,365 25-34 1904 1589 3494 35+ 1160 970 2630 9321 7317 16,639 A.) Explain why P(A)=0.44 B.) Find P(B). C.) Find the probability that the person chosen is a male that is at least 25 years old, P(A and B). Are the events A and B independent? A= {The person chosen is male} B={The person chosen is 25 years old or older}

Practice Problems Pg 430 #51 Most sample surveys use random digit dialing equipment to call residential telephone numbers at random. The telephone polling firm Zogby International reports that the probability that a call reaches a live person is 0.2. Calls are independent. a) A polling firm places 5 calls. What is the probability that none of them reaches a person? b) What is the probability that at least one of the 5 calls will reach a person?

Practice Problems: Pg 429 Ex 6.21 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 in a few minutes. Applied to people who have no HIV antibodies, one rapid test has a probability of 0.004 of producing a false positive. If a clinic tests 200 people who are free of HIV antibodies, what is the probability that at least one false-positive will occur?