Chapter1 - Course Notes

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Chapter 1 lecture notes Math 431, Spring 2010 Instructor: David F. Anderson Tentative Course Outline: I. Basic combinatorics: how to count. (a) No real probability yet. Just setting stage for how to calculate simple probabilities later. (b) Turns oﬀ a lot of students. Admittedly rather boring, but important. (c) One Week. II. Axioms of probability: building blocks of subject. (a) Basic question: what does it mean when we say something has a certain proba- bility? (b) Build up from basic axioms and prove basic (fundamental and useful) theorems. (c) One week. III. Conditional probability and independence: what does partial information get you (much more than that) (a) Hard. Maybe hardest of the whole semester conceptually. (b) Two weeks. IV. Everything else: Random variables, expectations, limit theorems,. .. (a) Functions of outcomes of experiments. Very important. This is what people want. (b) Can ask for probabilities, “expected values”, etc. (c) rest of course. 1

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The solution to many problems in probability are dependent upon being able to count the number of ways something can take place. Example: What is the probability of getting dealt a ﬂush in a game of poker? (A ﬂush is 5 cards of the same suit from a deck of 52—we are allowing straight ﬂushes also.) Solution: We can’t answer this yet, but what information do we need? How can we solve this problem? Suppose I tell you that there are m possible 5 card “hands” in poker. Suppose I also tell you that there are n possible hands that are ﬂushes. Then the probability of getting dealt a ﬂush should be # of possible hands that are ﬂushes # of possible hands = n m . Therefore, the problem is reduced to counting the number of ways to get both m and n . We’ll revisit this problem in a bit. Section 1.2: The basic principle of counting. Theorem (The basic principle of counting). Suppose that two experiments are to be per- formed. If experiment 1 can result in any one of m possible outcomes and if, for each outcome of experiment 1, there are n possible outcomes of experiment 2, then together there are mn possible outcomes of the two experiments. Example At a high school there are 12 teachers, with each teaching 4 courses. One teacher is to be given an award for best teaching for a given course . How many diﬀerent choices of teacher/course are possible? Solution: We regard the choice of teacher as the outcome of the ﬁrst experiment. The outcome of the second experiment is one of the four courses that teacher is teaching. From the basic principle of counting we see there are 12 × 4 = 48 possible choices. 2
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Chapter1 - Course Notes

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