Copyright 2013, 2010, 2007, Pearson, Education, Inc. Section 12.1 The Nature of Probability
Copyright 2013, 2010, 2007, Pearson, Education, Inc. What You Will Learn History The Nature of Probability Empirical Probability The Law of Large Numbers
Copyright 2013, 2010, 2007, Pearson, Education, Inc. History Probability is used in many areas, including public finance, medicine, insurance, elections, manufacturing, educational tests and measurements, genetics, weather forecasting, investments, opinion polls, the natural sciences, and games of chance
Copyright 2013, 2010, 2007, Pearson, Education, Inc. History Mathematical problems relating to games of chance were studied by a number of mathematicians of the Renaissance: Italy’s Girolamo Cardano (1501–1576) France’s Blaise Pascal (1623–1662) and Pierre de Fermat (1601–1665) Dutch Christian Huygens (1629–1695) Swiss Jacob Bernoulli (1654–1705) Pierre-Simon de Laplace (1749–1827)
Copyright 2013, 2010, 2007, Pearson, Education, Inc. Definitions An experiment is a controlled operation that yields a set of results. The possible results of an experiment are called its outcomes. An event is a subcollection of the outcomes of an experiment
Copyright 2013, 2010, 2007, Pearson, Education, Inc. Definitions Empirical probability is the relative frequency of occurrence of an event and is determined by actual observations of an experiment. Theoretical probability is determined through a study of the possible outcomes that can occur for the given experiment
Copyright 2013, 2010, 2007, Pearson, Education, Inc. Empirical Probability (Relative Frequency The empirical probability of an event, P(E), can be determined by the following formula
Copyright 2013, 2010, 2007, Pearson, Education, Inc. Example 1: Heads Up! In 100 tosses of a fair coin, 44 landed heads up. Determine the empirical probability of the coin landing heads up. Solution Let E be the event that the coin lands heads up
Copyright 2013, 2010, 2007, Pearson, Education, Inc. Empirical Probability in Genetics Using empirical probability, Gregor Mendel (1822–1884) developed the laws of heredity by crossbreeding different types of “pure” pea plants and observing the relative frequencies of the resulting offspring. These laws became the foundation for the study of genetics
Copyright 2013, 2010, 2007, Pearson, Education, Inc. Empirical Probability in Genetics
Copyright 2013, 2010, 2007, Pearson, Education, Inc. Empirical Probability in Genetics 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
Copyright 2013, 2010, 2007, Pearson, Education, Inc. Empirical Probability in Genetics
Copyright 2013, 2010, 2007, Pearson, Education, Inc. Empirical Probability in Genetics
Copyright 2013, 2010, 2007, Pearson, Education, Inc. Empirical Probability in Genetics Mendel concluded that the sex cells (now called gametes) of the pure yellow (dominant) pea plant carried some factor that caused the off-spring to be yellow and that the gametes of the green variety had a variant factor that “induced the development of green plants.” In 1909, Danish geneticist W. Johannsen called these factors “genes.”
Copyright 2013, 2010, 2007, Pearson, Education, Inc. The Law of Large Numbers The law of large numbers states that probability statements apply in practice to a large number of trials, not to a single trial. It is the relative frequency over the long run that is accurately predictable, not individual events or precise totals
Copyright 2013, 2010, 2007, Pearson, Education, Inc. Tossing a Fair Coin