Greenwood press
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book-20600
heart attack
no heart attack TOTAL aspirin 147 10,890 11,037 placebo 146 10,888 11,034 TOTAL 293 21,778 22,071 Expected frequencies for the aspirin–physician heart study The computations assume that the totals represent the population, and that heart condition is inde- pendent of medication. The statistician conducts a chi-square test to compare the actual frequencies to the expected frequencies. In this case, the chi-square indicated that the ob- 78 PROBABILITY served frequencies were not close to the expected values, so aspirin reduced heart attacks. Making careful lists and working from simple examples can determine many probability problems. How many families with three children have exactly two boys? If boys and girls are equally likely, you can list eight possibilities: BBB, BBG, BGB, BGG, GBB, GBG, GGB, and GGG. The list is called the sample space, because each family is equally likely. Three of these, BBG, BGB, and GBB, represent two boys and one girl. So the probability of a family of three chil- dren having exactly two boys is three-eights, or 37.5 percent. The problem of finding how many families would have two boys in three children can be approached through a simulation. A simulation replaces the ele- ments of this problem with repeated trials of an experiment using objects that behave like the birth of children. Tossing a coin could represent the birth of a child. If you were to determine boys by the head of the coin showing, you could simulate a family of three children by tossing three coins, say a penny for the first child, a dime for the second child, and a quarter for the third. This experiment can be carried 500 or more times very quickly. The probability of two boys would be estimated by the proportion of times the three coins showed exactly two heads. In one experiment, this proportion turned out to be 35.8 percent, which is a little less than the value computed from the sample space. It is now common to use computers to model complex relationships with simulations. Computers can generate random numbers (or numbers that act randomly) and perform rapid computation of probabilities. The Defense Department uses simu- lations to evaluate outcomes of military actions. Aircraft designers use computer simulations of air molecules hitting the surface of an airplane to determine its most efficient shape. The Centers for Disease Control uses simulations to predict the paths of epidemics. It makes recommendations for vaccinations and preven- tion procedures based on the outcomes of its simulations. Coins and children present examples of binomial probability situations. When there are two outcomes of a single trial (heads or tails on one coin, boy or girl for one birth), and a fixed number of independent trials, the computation of outcome probabilities can be generated by terms in the expansion of the binomial (p + q) n , where n is the number of trials, p is the probability of one outcome (called the success), and q = 1 − p is the probability of failure. Families of three children would be modeled by (p + q) 3 = p 3 + 3p 2 q + 3pq 2 + q 3 . The term 3p 2 q would represent the probability of two boys and one girl. Since p = q = 1 2 , the value 3p 2 q = 3 8 agrees with our previous computation. The binomial probability theorem provides direct solutions for problems that don’t have equal probabilities such as the proportion of recessive genes in a pop- ulation or how many people should be booked for flights so that there are no empty seats. In a situation in which there are different percentages of a dominant gene A and a recessive gene a, shouldn’t the dominant gene eventually “win out” Download 1.81 Mb. Do'stlaringiz bilan baham: |
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