Bibliography. The Drunkard s Walk: How Randomness Rules Our Lives by Leonard Mlodinow. Duelling Idiots and Other Probability Puzzlers by Paul J.

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3 Bibliography The Drunkard s Walk: How Randomness Rules Our Lives by Leonard Mlodinow Duelling Idiots and Other Probability Puzzlers by Paul J. Nahin The History of mathematics by David M. Burton Is God a Mathematician? by Mario Livio Mathematical Analysis by Arya and Lardner Mathematical Ideas by Miller, Heeren & Hornsby

4 Introduction to Probability The Romans generally scorned mathematics. In the words of the Roman statesman Cicero ( BCE), The Greeks held the geometer in the highest honor; accordingly, nothing made more brilliant progress among them but mathematics. But we have established as the limit in that art its usefulness in measuring and counting. Indeed, whereas one might imagine a Greek textbook focused on the proof of congruences of abstract triangles, a typical Roman text focused on such issues as how to determine the width of a river when the enemy is occupying the other bank. In Roman culture it was comfort and war, not truth and beauty, that occupied center stage. because they focused on the practical, they did find value in probability; Cicero wrote probability is the very guide of life. [ Mlodinow, page 31.]

5 Obvious Probability. 1. A fair coin is flipped 3 times. If all three are heads, what is the probability of getting a heads on the fourth flip? 2. What is the probability of rolling a fair die and having a 6 show up? 3. In a family with a husband, a wife, and two children, what is the probability that the children are both girls? 4. In 1992 there were 4 million women in the U.S. who were battered by their husbands or boyfriends. In that same year, a total of 1,432 of those women were murdered. What is the probability that a battered women was killed by her husband or boyfriend?

6 Less Obvious Probability 1. A fair coin is flipped two times. The number of heads that can turn up is 0,1, or 2. What is the probability of getting 0 heads? 2. A family has two children and you know that one is a girl. What is the probability that both are girls? 3. In the above family, the girl you know is named Ruth. What is the probability that both are girls? 4. Three prisoners, A,B, and C are up for parole and each has an equal chance but only one of them will get it. In a conversation with the warden, A is told that now prisoner B, because of something he did, will definitely not get paroled. Prisoner A is excited by this and tells C the news. What is the probability that C will get paroled?

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8 Blaise Pascal

9 Set Theory A Set is a collection of objects. The objects are called elements of the set. Particular sets can be defined by (i) a word description or (ii) a list of its elements or (iii) the use of set-builder notation. Example: (i) The set of counting numbers from 1 to 10 (ii) {1,2,3,4,5,6,7,8,9,10} or {1,2,3,...,10} (iii) { n n is a counting number from 1 to 10 } For a set to be useful, it must be well defined; that is, given any object, we must be able to tell whether or not it is an element of the given set.

10 An important set is the one with no elements; we call it the empty or null set and denote it by { } or by the symbol. The empty set can be described by words: The set of all green elephants in this room, or by a listing: { }, or by a set builder: {n n is a counting number between 3 and 4}. SYMBOLS: Sets are usually designated by capital letters and their elements by small letters. For example, Let A = {1,2,3} and B = {1,2,3,4,5} and C = {1,2,3,...}. To indicate an element of a set, we use the symbol. Thus, we can write 2 A or write n B to indicate that n is one of the numbers from 1 to 5. Note that the elements of C go on forever. To indicate an object is not an element of a set, we use. For example, 6 B and for any x whatsoever, x { }.

11 Venn Diagram U A A B C U is called the universal set A is called the complement of A and A = {x x A} We can observe that every element of B is also an element of A. In this case, we say B is a subset of A or B is contained in A and write B A. Is C a subset of A? Note: For any set A, A and A A

12 U A A C A union C is a new set: A C = {x x A or x C} Note: Union is like addition; and for any set A, A =A A intersect C is a new set; A C = {x x A and x C} A minus C is a new set; A C = {x x A and x C}

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14 Now let s get back to the notion of a sample space. DEFINITIONS. RANDOM: non predictable. Experiment: observation of a random, repeatable happening. Outcome: the results of an experiment. Sample Space: a set S of all possible outcomes. Event: any subset of the sample space. Elements of an event are called successes.

15 Definition: To each outcome x in a sample space S, we assign a unique number P(x) (read p of x ) called the probability of x happening. P has the following properties: 1. 0 P(x) 1 for all x in S 2. P( ) = 0 3. The sum of the probabilities of all x in S is 1. For any event A S, P(A) is the probability that the event A happens and is the sum of P(x) for all outcomes x in A. Note: P(S) = 1 Important: (1) P(A ) = 1 - P(A) and (2) If the probability is uniform (each outcome has the same probability), then P(A) = the number of outcomes in A divided by the total number of outcomes.

16 Counting Principal: Suppose that a random process can have m outcomes and another random process can have n outcomes. Then the number of possible outcomes (i.e., the number of points in the sample space) of performing the processes in sequence is mn. This can be extended to any number of processes. Examples: (1) Toss a fair coin two times. (2) Family with three children. (3) Roll a pair of dice. (4) Deal out 5 cards from a 52 card deck. (311,875,200 points in the sample space.) Note: These are finite sample spaces. A slightly different case: (5) Toss a coin until the first head appears. This sample space is infinite.

17 1. A fair coin is flipped 3 times. If all three are heads, what is the probability of getting a heads on the fourth flip? What is a sample space for this problem? 2. What is the probability of rolling a fair die and having a 6 show up? What is a sample space for this problem? 3. In a family with a husband, a wife, and two children, what is the probability that the children are both girls? 4. In 1992 there were 4 million women in the U.S. who were battered by their husbands or boyfriends. In that same year, a total of 1,432 of those women were murdered. What is the probability that a battered women was killed by her husband or boyfriend?

18 . A fair coin is flipped two times. The number of heads that can turn up is 0,1, or 2. What is the probability of getting 0 heads? 2. A family has two children and you know that one is a girl. What is the probability that both are girls? 3. In the above family, the girl you know is named Ruth. What is the probability that both are girls? 4. Three prisoners, A,B, and C are up for parole and each has an equal chance but only one of them will get it. In a conversation with the warden, A is told that prisoner B will definitely not get paroled. Prisoner A is excited by this and tells C the news. What is the probability that C will get paroled?

19 No class next week