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UT Knoxville STAT 201 - Chapter 14

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1Chapter14 Presentation 1213Copyright © 2009 Pearson Education, Inc.Chapter 14 From Randomness to ProbabilityChapter14 Presentation 1213Copyright © 2009 Pearson Education, Inc.2Gambling - The Birth of Statistics If I flipped a coin and got 4 heads in a row, would you: bet money that tails will be next, because it’s “due”? bet money that heads will be next, because the coin is “on a streak”? conclude that my coin is unfair, not yielding 50/50 results? How unusual is 4 heads in a row?Chapter14 Presentation 1213Copyright © 2009 Pearson Education, Inc.3Dealing with Random Phenomena What do we mean by a “random phenomenon”? 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.Chapter14 Presentation 1213Copyright © 2009 Pearson Education, Inc.4First a definition . . . When thinking about what happens with combinations of outcomes (i.e., “events”), 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.The Law of Large NumbersChapter14 Presentation 1213Copyright © 2009 Pearson Education, Inc.5The 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.Chapter14 Presentation 1213Copyright © 2009 Pearson Education, Inc.6The Law of Large Numbers ExampleThe overall percentage of times the light is greensettles down as you see more outcomes.Probability of green light looks to be about 0.30Chapter14 Presentation 1213Copyright © 2009 Pearson Education, Inc.7The “Law of Large Numbers” vs. the “Law of Averages” The “Law of Large Numbers” says relative frequencies even out only in the long run, and this long run is really long (infinitely long, in fact). The “Law of Averages” says (for example) that if you flip a fair coin 4 times and get heads all 4 times, tails are “due” on the 5thflip. If it’s really a “fair” coin, is this true?Chapter14 Presentation 1213Copyright © 2009 Pearson Education, Inc.8Law of Large Numbers and Coin Flips Careful: the law of large numbers talks about the relative frequency of an event, not the number of times the event happens. Notice that the difference in the # of H vs. T (top picture) isnot approaching 0 as the number of tosses increases, but the relative frequency of H (bottom picture) is approaching 0.50Relative Frequency of HeadsChapter14 Presentation 1213Copyright © 2009 Pearson Education, Inc.9Mathematically Determined Probabilities In some situations, repeated observation is not necessary to know the probability of an event. The long run relative frequency can be mathematically determined. All gambling games of chance are examples.Chapter14 Presentation 1213Copyright © 2009 Pearson Education, Inc.10Personal Probability In everyday speech, when we express a degree of uncertainty without basing it on long-run relative frequencies or mathematical calculations, we are stating subjective or personalprobabilities. Personal probabilities don’t display the kind of consistency that we will need probabilities to have, so we’ll stick with formally defined probabilities.Chapter14 Presentation 1213Copyright © 2009 Pearson Education, Inc.11Modeling Probability Mathematically calculating the probability of an event is simplified when all the possible outcomes are equally likely. However, keep in mind that all possible outcomes are not always equally likely.Chapter14 Presentation 1213Copyright © 2009 Pearson Education, Inc.12 If each possible outcome is equally likely, the probability of an event is the number of outcomes in the event divided by the total number of possible outcomes.P(A) = Modeling Probability (cont.)# of outcomes in A# of possible outcomesChapter14 Presentation 1213Copyright © 2009 Pearson Education, Inc.13Modeling Probability - Example Flip a fair coin 4 times. There are 16 possible equally likely outcomes: How would you interpret this probability?If event A is getting at least 3 heads, what is P(A)?Chapter14 Presentation 1213Copyright © 2009 Pearson Education, Inc.14Formal Probability1. Requirement for a probability: A probability is a number between 0 and 1.  For any event A, 0 ≤ P(A) ≤ 1. If P(A)=0, event A can never happen.  If P(A)=1, event A will happen, with absolutecertainty.Chapter14 Presentation 1213Copyright © 2009 Pearson Education, Inc.15Formal Probability (cont.)2. Probability Assignment Rule: Let S represents the set of all possible outcomes. P(S) = 1 Example: roll a fair six-sided die, record the number of dots on the up face.S = {1, 2, 3, 4, 5, 6}P(S) = probability of getting a 1 or a 2 or a 3 or a 4 or a 5 or a 6 = 1.00Chapter14 Presentation 1213Copyright © 2009 Pearson Education, Inc.16Formal Probability (cont.)3. Complement Rule: The set of outcomes that are not in the event Ais called the complement of A. The probability of an event occurring is 1 minus the probability that it doesn’t occur:P(A) = 1 – P(not A)Chapter14 Presentation 1213Copyright © 2009 Pearson Education, Inc.17Compliment Rule (cont.) Examples where it’s easier to calculate the probability of the event not happening: What’s the probability that, in a room of 50 people, one or more people share the same birthday? What’s the probability at least one other person on board a plane is a certified pilot if the original and copilot dies?Chapter14 Presentation 1213Copyright © 2009 Pearson Education, Inc.18Formal Probability (cont.)4. Addition Rule: Events that have no outcomes in common (and, thus, cannot occur simultaneously) are called disjoint (or mutually exclusive). Example: Roll a single die.Event A = roll a 5 or greaterEvent B = roll a 2 or lessEvent C = roll an even number Events A and B are disjoint: they can’t happen on a single roll. Events A and C are not disjoint, they could both happen on a single roll (when you roll a 6).Chapter14 Presentation 1213Copyright © 2009


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UT Knoxville STAT 201 - Chapter 14

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