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Chapter 14 From Randomness to Probability Copyright © 2009 Pearson Education, Inc. Dealing with Random Phenomena A random phenomenon is a situation in which we know what outcomes could happen, but we don’t know which particular outcome did or will happen. In general, each occasion upon which we observe a random phenomenon is called a trial. At each trial, we note the value of the random phenomenon, and call it an outcome. When we combine outcomes, the resulting combination is an event. The collection of all possible outcomes is called the sample space. Copyright © 2009 Pearson Education, Inc. Slide 1- 2 The Law of Large Numbers First a definition . . . When thinking about what happens with combinations of outcomes, things are simplified if the individual trials are independent. Roughly speaking, this means that the outcome of one trial doesn’t influence or change the outcome of another. For example, coin flips are independent. Copyright © 2009 Pearson Education, Inc. Slide 1- 3 The Law of Large Numbers (cont.) The Law of Large Numbers (LLN) says that the long-run relative frequency of repeated independent events gets closer and closer to a single value. We call the single value the probability of the event. Because this definition is based on repeatedly observing the event’s outcome, this definition of probability is often called empirical probability. Copyright © 2009 Pearson Education, Inc. Slide 1- 4 Foundation of Probability The onset of probability as a useful science is primarily attributed to Blaise Pascal (1623-1662) and Pierre de Fermat (1601-1665). While contemplating a gambling problem posed by Chevalier de Mere in 1654, Blaise Pascal and Pierre de Fermat laid the fundamental groundwork of probability theory, and are thereby accredited the fathers of probability. Chances of a Lifetime Copyright © 2009 Pearson Education, Inc. Slide 1- 5 Modeling Probability (cont.) The probability of an event is the number of outcomes in the event divided by the total number of possible outcomes. P(A) = # of outcomes in A # of possible outcomes Copyright © 2009 Pearson Education, Inc. Slide 1- 6 The First Three Rules of Working with Probability (MAKE A PICTURE) The most common kind of picture to make is called a Venn diagram. We will see Venn diagrams in practice shortly… Copyright © 2009 Pearson Education, Inc. Slide 1- 7 Formal Probability 1. Two requirements for a probability: A probability is a number between 0 and 1. For any event A, 0 ≤ P(A) ≤ 1. 2. Probability Assignment Rule: The probability of the set of all possible outcomes of a trial must be 1. P(S) = 1 (S represents the set of all possible outcomes.) Copyright © 2009 Pearson Education, Inc. Slide 1- 8 Formal Probability (cont.) 3. Complement Rule: The set of outcomes that are not in the event A is called the complement of A, denoted AC. The probability of an event occurring is 1 minus the probability that it doesn’t occur: P(A) = 1 – P(AC) Copyright © 2009 Pearson Education, Inc. Slide 1- 9 Formal Probability (cont.) 4. Addition Rule: Events that have no outcomes in common (and, thus, cannot occur together) are called disjoint (or mutually exclusive). Copyright © 2009 Pearson Education, Inc. Slide 1- 10 Formal Probability (cont.) 4. Addition Rule (cont.): For two disjoint events A and B, the probability that one or the other occurs is the sum of the probabilities of the two events. P(A or B) = P(A) + P(B), provided that A and B are disjoint. Copyright © 2009 Pearson Education, Inc. Slide 1- 11 Formal Probability 5. Multiplication Rule: For two independent events A and B, the probability that both A and B occur is the product of the probabilities of the two events. P(A and B) = P(A) x P(B), provided that A and B are independent. Copyright © 2009 Pearson Education, Inc. Slide 1- 12 Formal Probability (cont.) 5. Multiplication Rule (cont.): Two independent events A and B are not disjoint, provided the two events have probabilities greater than zero: Copyright © 2009 Pearson Education, Inc. Slide 1- 13 Formal Probability (cont.) 5. Multiplication Rule: Many Statistics methods require an Independence Assumption, but assuming independence doesn’t make it true. Always Think about whether that assumption is reasonable before using the Multiplication Rule. Copyright © 2009 Pearson Education, Inc. Slide 1- 14 Formal Probability - Notation Notation: In this text we use the notation P(A or B) and P(A and B). In other situations, you might see the following: P(A B) instead of P(A or B) P(A B) instead of P(A and B) Copyright © 2009 Pearson Education, Inc. Slide 1- 15 Example #1 A survey of 64 informed voters revealed the following information: 45 believe that Elvis is still alive 49 believe that they have been abducted by space aliens 42 believe both of these things Copyright © 2009 Pearson Education, Inc. Slide 1- 16 Example #2 A survey of 88 faculty and graduate students at the University of Florida's film school revealed the following information: 51 admire Moe 49 admire Larry 60 admire Curly 34 admire Moe and Larry 32 admire Larry and Curly 36 admire Moe and Curly 24 admire all three of the Stooges Copyright © 2009 Pearson Education, Inc. Slide 1- 17 Chapter 15 Probability Rules! Copyright © 2009 Pearson Education, Inc. The General Addition Rule When two events A and B are disjoint, we can use the addition rule for disjoint events from Chapter 14: P(A or B) = P(A) + P(B) However, when our events are not disjoint, this earlier addition rule will double count the probability of both A and B occurring. Thus, we need the General Addition Rule. Copyright © 2009 Pearson Education, Inc. Slide 1- 19 General Addition Rule: For any two events A and B, P(A or B) = P(A) + P(B) – P(A and B) (On the Formula Sheet) The following Venn diagram shows a situation in which we would use the general addition rule: Copyright © 2009 Pearson Education, Inc. Slide 1- 20 Conditional Probabilities To find the probability of the event B given the event A, we restrict our attention to the outcomes in A. We then find in what fraction of those outcomes B also occurred. P (A and B) P(B|A) P(A) (On the Formula Sheet) Note: P(A) cannot equal 0, since we know that A has occurred. Copyright © 2009 Pearson Education, Inc. Slide 1- 21 The General Multiplication Rule When two events A and B are independent, we can use the multiplication rule for independent events: P(A and B) = P(A) x P(B) However, when our events are not independent, this earlier multiplication rule does not work. Thus, we need the General Multiplication Rule. Copyright © 2009 Pearson Education, Inc. Slide 1- 22 The General Multiplication Rule (cont.) We encountered the general multiplication rule in the form of conditional probability. Rearranging the equation in the definition for conditional probability, we get the General Multiplication Rule: For any two events A and B, P(A and B) = P(A) x P(B|A) or P(A and B) = P(B) x P(A|B) Copyright © 2009 Pearson Education, Inc. Slide 1- 23 Independence Independence of two events means that the outcome of one event does not influence the probability of the other. With our new notation for conditional probabilities, we can now formalize this definition: Events A and B are independent whenever P(B|A) = P(B). (Equivalently, events A and B are independent whenever P(A|B) = P(A).) Copyright © 2009 Pearson Education, Inc. Slide 1- 24 Independent ≠ Disjoint Disjoint events cannot be independent! Well, why not? Since we know that disjoint events have no outcomes in common, knowing that one occurred means the other didn’t. Thus, the probability of the second occurring changed based on our knowledge that the first occurred. It follows, then, that the two events are not independent. A common error is to treat disjoint events as if they were independent, and apply the Multiplication Rule for independent events—don’t make that mistake. Copyright © 2009 Pearson Education, Inc. Slide 1- 25 Drawing Without Replacement Sampling without replacement means that once one object is drawn it doesn’t go back into the pool. We often sample without replacement, which doesn’t matter too much when we are dealing with a large population. However, when drawing from a small population, we need to take note and adjust probabilities accordingly. Drawing without replacement is just another instance of working with conditional probabilities. Copyright © 2009 Pearson Education, Inc. Slide 1- 26 Reversing the Conditioning Reversing the conditioning of two events is rarely intuitive. Suppose we want to know P(A|B), but we know only P(A), P(B), and P(B|A). We also know P(A and B), since P(A and B) = P(A) x P(B|A) From this information, we can find P(A|B): P (A and B) P(A|B) P(B) Copyright © 2009 Pearson Education, Inc. Slide 1- 27 Chapter 16 Random Variables Copyright © 2009 Pearson Education, Inc. Expected Value: Center A random variable assumes a value based on the outcome of a random event. We use a capital letter, like X, to denote a random variable. A particular value of a random variable will be denoted with a lower case letter, in this case x. Copyright © 2009 Pearson Education, Inc. Slide 1- 29 Expected Value: Center (cont.) There are two types of random variables: Discrete random variables can take one of a finite number of distinct outcomes. Example: Number of credit hours Continuous random variables can take any numeric value within a range of values. Example: Cost of books this term Copyright © 2009 Pearson Education, Inc. Slide 1- 30 Expected Value: Center (cont.) A probability model for a random variable consists of: The collection of all possible values of a random variable, and the probabilities that the values occur. Of particular interest is the value we expect a random variable to take on, notated μ (for population mean) or E(X) for expected value. Copyright © 2009 Pearson Education, Inc. Slide 1- 31 Expected Value: Center (cont.) The expected value of a (discrete) random variable can be found by summing the products of each possible value and the probability that it occurs: E X x P x Note: Be sure that every possible outcome is included in the sum and verify that you have a valid probability model to start with. Copyright © 2009 Pearson Education, Inc. Slide 1- 32 First Center, Now Spread… For data, we calculated the standard deviation by first computing the deviation from the mean and squaring it. We do that with discrete random variables as well. The variance for a random variable is: Var X x P x 2 2 The standard deviation for a random variable is: SD X Var X Copyright © 2009 Pearson Education, Inc. Slide 1- 33 Continuous Random Variables Random variables that can take on any value in a range of values are called continuous random variables. Continuous random variables have means (expected values) and variances. We won’t worry about how to calculate these means and variances in this course, but we can still work with models for continuous random variables when we’re given these parameters. Copyright © 2009 Pearson Education, Inc. Slide 1- 34