11.1 Empirical and Theoretical Probabilities 661 Empirical Probability in Genetics Using empirical probability, Gregor Mendel (1822–1884) developed the laws of heredity by crossbreeding different types of pea plants and observing the relative frequencies of the resulting offspring. These laws became the foundation for the study of genetics. In plant genetics, a true-breeding or a true plant always passes down certain traits to its offspring. For example, a true yellow-pea plant when crossbred with another true yellow-pea plant would always produce another true yellow-pea plant. When Mendel crossbred a true yellow-pea plant and a true green-pea plant, the resulting offspring (the first generation) were always yellow; see Fig. 11.1(a). When he crossbred a true round-seeded pea plant and a true wrinkled-seeded pea plant, the resulting offspring (the first generation) were always round; see Fig. 11.1(b). True yellow True green True round True wrinkled Yellow (not true) Round (not true) Firstgeneration offspring Yellow – dominant, green – recessive (a) Round – dominant, wrinkled – recessive (b) Figure 11.1 MATHEMATICS TODAY Genetic Conditions Seeing a dog or a person with different-colored eyes is unusual, but not that unusual. Features such as eye color are determined genetically, as are certain illnesses. Below is a chart listing some human genetic conditions and their rates of occurrence in the United States. Condition/Incidence Cystic fibrosis: 1 out of 3000 white births Down syndrome: 1 out of 700 births Duchenne muscular dystrophy: 1 out of 3600 male births Fragile X syndrome: 1 out of 4000 male births 1 out of 7000 female births Hemophilia A: 1 out of 5600 male births Polycystic kidney disease: 1 out of 2500 births Prader–Willi syndrome: 1 out of 20,000 births Sickle-cell anemia: 1 out of 400 African American births Tay-Sachs disease: 1 out of 3500 Ashkenazi Jewish births Why This Is Important Genetics, which has its foundation in probability, is used in explaining the cause of these illnesses, and it plays an important role in research to discover treatments for these and other genetic conditions. Mendel called traits such as yellow color and round seeds dominant because they overcame or “dominated” the other trait. He labeled the green color and the wrinkled traits recessive . Mendel then crossbred the offspring of the first generation. The resulting secondgeneration offspring had both the dominant and the recessive traits of their grandparents; see Fig. 11.2(a) and 11.2(b). What’s more, these traits always appeared in approximately a 3 to 1 ratio of dominant to recessive. First-generation offspring Second-generation offspring Yellow (not true) Yellow (not true) Yellow or green Second-generation offspring – approximately 3 yellow to every 1 green (a) Round (not true) Round (not true) Round or wrinkled Second-generation offspring – approximately 3 round to every 1 wrinkled (b) Figure 11.2 Table 11.2 lists some of the actual results of Mendel’s experiments with pea plants. Note that the ratio of dominant trait to recessive trait in the second-generation offspring is about 3 to 1 for each experiment. The empirical probability of the dominant trait has also been calculated. How would you determine the empirical probability of the recessive trait?
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