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Basic Concepts of Probability MATH 100, Survey of Mathematical Ideas J. Robert Buchanan Department of Mathematics Fall 2014 J. Robert Buchanan Basic Concepts of Probability Background The early development of the theory of probability was driven by gambling. Some occurrences are deterministic (drop a book, it will hit the floor). Other occurrences are random (flip a fair coin, the outcome will be heads half the time). Probability is the branch of mathematics dedicated to determining the likelihood of random phenomena. Any observation of measurement of a random phenomena is called an experiment. The results of an experiment are called outcomes. The set of all possible outcomes of an experiment is called the sample space. J. Robert Buchanan Basic Concepts of Probability Theoretical Probability Formula If all the outcomes in a sample space S are equally likely, and E is an event within the sample space, then the theoretical probability of event E is P(E) = number of favorable outcomes n(E) = total number of outcomes n(S) J. Robert Buchanan Basic Concepts of Probability Example A group of preschool children includes eight girls and five boys. One child from the group is randomly selected, with E being the outcome that the selected child is a girl. What are the following? 1 The total number of outcomes? 2 The number of favorable outcomes? 3 The number of unfavorable outcomes? 4 The probability of event E? J. Robert Buchanan Basic Concepts of Probability Empirical Probability Formula Sometimes we do not know that all events in the sample space occur with equal probability. In this case we may perform an experiment to approximate the probability of an outcome. J. Robert Buchanan Basic Concepts of Probability Empirical Probability Formula Sometimes we do not know that all events in the sample space occur with equal probability. In this case we may perform an experiment to approximate the probability of an outcome. If E is an event that may happen when an experiment is performed, then the empirical probability of event E is P(E) ≈ number of times event E occurred . number of times the experiment was performed J. Robert Buchanan Basic Concepts of Probability Empirical Probability Formula Sometimes we do not know that all events in the sample space occur with equal probability. In this case we may perform an experiment to approximate the probability of an outcome. If E is an event that may happen when an experiment is performed, then the empirical probability of event E is P(E) ≈ number of times event E occurred . number of times the experiment was performed Example When 200 seeds were planted, 170 of them germinated. Find the probability that this type of seed will germinate. J. Robert Buchanan Basic Concepts of Probability Law of Large Numbers As an experiment is repeated more and more times, the proportion of outcomes favorable to any particular event will tend to come closer and closer to the theoretical probability of that event. J. Robert Buchanan Basic Concepts of Probability Example Suppose three fair coins are tossed. 1 What is the sample space? 2 What is the probability of no heads? 3 What is the probability of exactly one head? 4 What is the probability of exactly two heads? 5 What is the probability of three heads? J. Robert Buchanan Basic Concepts of Probability Example A family would like to have exactly two daughters. What is the probability of having exactly two daughters, if the family has four children all together? J. Robert Buchanan Basic Concepts of Probability Probability in Genetics (1 of 2) The father of modern genetics (Gregor Mendel) observed the following inheritance behavior in pea plants. Pure red plants crossed with pure white plants produce red plants. J. Robert Buchanan Basic Concepts of Probability Probability in Genetics (1 of 2) The father of modern genetics (Gregor Mendel) observed the following inheritance behavior in pea plants. Pure red plants crossed with pure white plants produce red plants. First to Second Generation Second Parent r r First R Rr Rr Parent R Rr Rr Each member of the second generation is red, since red is the dominant gene. J. Robert Buchanan Basic Concepts of Probability Probability in Genetics (2 of 2) Suppose the plants from the second generation are crossed with each other. Second to Third Generation Second Parent R r First R RR Rr Parent r rR rr J. Robert Buchanan Basic Concepts of Probability Probability in Genetics (2 of 2) Suppose the plants from the second generation are crossed with each other. Second to Third Generation Second Parent R r First R RR Rr Parent r rR rr 1 What is the probability that a plant in the third generation is red? 2 What is the probability that a plant in the third generation is white? J. Robert Buchanan Basic Concepts of Probability Odds Recall: the probability of an event compares favorable outcomes to the total number of possible outcomes. In gambling the number of favorable events is often compared to the number of unfavorable events. J. Robert Buchanan Basic Concepts of Probability Odds Recall: the probability of an event compares favorable outcomes to the total number of possible outcomes. In gambling the number of favorable events is often compared to the number of unfavorable events. Definition If all outcomes in a sample space are equally likely, with a of them favorable to event E and the remaining b outcomes unfavorable to event E, then the odds in favor of E are a to b, and the odds against E are b to a. J. Robert Buchanan Basic Concepts of Probability Example A jar contains 5 red balls, 6 yellow balls, and 3 green balls. A blind-folded person will randomly select one ball from the jar. 1 Find the odds in favor of selecting a red ball. 2 Find the odds against selecting a yellow ball. 3 Find the odds in favor of selecting a green ball. J. Robert Buchanan Basic Concepts of Probability Converting Between Odds and Probabilities If the odds in favor of event E are a to b, then P(E) = J. Robert Buchanan a . a+b Basic Concepts of Probability Converting Between Odds and Probabilities If the odds in favor of event E are a to b, then P(E) = a . a+b Example Find the probability of winning a lottery if the odds in favor of wining are 1 to 125, 000. J. Robert Buchanan Basic Concepts of Probability Example Suppose the probability of event E is P(E) = 0.124, what are the odds in favor of event E? J. Robert Buchanan Basic Concepts of Probability Example Suppose the probability of event E is P(E) = 0.124, what are the odds in favor of event E? P(E) = 0.124 = 31 31 124 = = 1000 250 31 + 219 The odds in favor of E are 31 to 250 − 31 = 219. J. Robert Buchanan Basic Concepts of Probability Homework Read Section 12.1. Exercises: 1, 5, 9, 13, . . . , 45, 49 J. Robert Buchanan Basic Concepts of Probability