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Lecture Slides Elementary Statistics Twelfth Edition and the Triola Statistics Series by Mario F. Triola Copyright © 2014, 2012, 2010 Pearson Education, Inc. Section 4.2-1 Chapter 4 Probability 4-1 Review and Preview 4-2 Basic Concepts of Probability 4-3 Addition Rule 4-4 Multiplication Rule: Basics 4-5 Multiplication Rule: Complements and Conditional Probability 4-6 Counting 4-7 Probabilities Through Simulations 4-8 Bayes’ Theorem Copyright © 2014, 2012, 2010 Pearson Education, Inc. Section 4.2-2 Key Concept This section presents three approaches to finding the probability of an event. The most important objective of this section is to learn how to interpret probability values. Copyright © 2014, 2012, 2010 Pearson Education, Inc. Section 4.2-3 Definitions Event any collection of results or outcomes of a procedure Simple Event an outcome or an event that cannot be further broken down into simpler components Sample Space for a procedure consists of all possible simple events; that is, the sample space consists of all outcomes that cannot be broken down any further Copyright © 2014, 2012, 2010 Pearson Education, Inc. Section 4.2-4 Example In the following display, we use “b” to denote a baby boy and “g” to denote a baby girl. Procedure Example of Event Sample Space Single birth 1 girl (simple event) {b, g} 3 births 2 boys and 1 girl (bbg, bgb, and gbb are all simple events) {bbb, bbg, bgb, bgg, gbb, gbg, ggb, ggg} Copyright © 2014, 2012, 2010 Pearson Education, Inc. Section 4.2-5 Notation for Probabilities P - denotes a probability. A, B, and C - denote specific events. P(A) - denotes the probability of event A occurring. Copyright © 2014, 2012, 2010 Pearson Education, Inc. Section 4.2-6 Basic Rules for Computing Probability Rule 1: Relative Frequency Approximation of Probability Conduct (or observe) a procedure, and count the number of times event A actually occurs. Based on these actual results, P(A) is approximated as follows: P(A) = # of times A occurred # of times procedure was repeated Copyright © 2014, 2012, 2010 Pearson Education, Inc. Section 4.2-7 Example – Relative Frequency Concept of Probability Example #9: A sample of 820 adults showed that 80 of them had no credit cards, 116 had one card each, 94 had two cards each, 77 had three cards each, 43 had four cards each, and 410 had five or more cards each. Write the frequency distribution table for the number of credit cards an adult possesses. Calculate the relative frequencies for all categories. Suppose one adult is randomly selected from these 820 adults. Find the probability that this adult has (a) three credit cards (b) five or more cards Solution: # of Credit Cards 0 1 2 3 4 >=5 Frequency (f) 80 116 94 77 43 410 ∑f = 820 Relative Frequency (rf) 80/820 = 0.0976 116/820 = 0.1415 94/820 = 0.1146 77/820 = 0.0939 43/820 = 0.0524 410/820 = 0.5000 ∑rf = 1.0000 a. P(3 cards) = 0.0939 b. P(>=5) = 0.5000 Copyright © 2014, 2012, 2010 Pearson Education, Inc. Section 4.2-8 Basic Rules for Computing Probability Rule 2: Classical Approach to Probability (Requires Equally Likely Outcomes) Assume that a given procedure has n different simple events and that each of those simple events has an equal chance of occurring. If event A can occur in s of these n ways, then s number of ways A can occur P ( A) = = n number of different simple events Copyright © 2014, 2012, 2010 Pearson Education, Inc. Section 4.2-9 Example – Classical Probability Example #6: Solution: A box contains 40 marbles. Of them, 18 are red and 22 are green. If one # of events favorable to red 18 a. P ( red ) 0.45 marble is randomly selected out of this Total # of outcomes 40 box, what is the probability that this # of events favorable to green 22 marble is: b. P(green ) 0.55 Total # of outcomes 40 a. red? b. green? Example #7 A multiple-choice question on a test has five answers. If Dianne chooses one answer based on “pure guess” , what is the probability that her answer is: a. correct? b. wrong? Do these two probabilities add up to 1? If yes why? Solution: # of events favorable to correct 1 0.2 Total # of outcomes 5 # of events favorable to wrong 4 b. P( wrong ) 0.8 Total # of outcomes 5 a. P(correct ) Yes. The experiment has two and only two outcomes and according to the 2nd property of probability, the sum of the probability must be 1. Copyright © 2014, 2012, 2010 Pearson Education, Inc. Section 4.2-10 Example – Classical Probability Example #5: Suppose a randomly selected passenger is about to go through the metal detector at JFK Airport in New York City. Consider the following two outcomes. The passenger sets off the metal detector, or the passenger does not set off the metal detector. Are the two outcomes equally likely? Explain why or why not. If you are to find the probability of these two outcomes, would you use the classical approach or another approach? Explain why. Solution: a. The two outcomes, “passenger sets off the metal detector” and “passenger does not set of the metal detector”, are not equally likely because if they were, 50% of the passengers would set off the detector. This would be a daunting task for the Transportation Authority Administration (TSA). b.Classical approach will not be appropriate for determining the probability of these two outcomes. Therefore, another approach is needed. This other approach will require obtaining a random sample of passengers going through NY JFK airport and collecting information whether they set off the detector or not. Copyright © 2014, 2012, 2010 Pearson Education, Inc. Section 4.2-11 Basic Rules for Computing Probability Rule 3: Subjective Probabilities P(A), the probability of event A, is estimated by using knowledge of the relevant circumstances. Copyright © 2014, 2012, 2010 Pearson Education, Inc. Section 4.2-12 Law of Large Numbers As a procedure is repeated again and again, the relative frequency probability of an event tends to approach the actual probability. Copyright © 2014, 2012, 2010 Pearson Education, Inc. Section 4.2-13 Example When three children are born, the sample space is: {bbb, bbg, bgb, bgg, gbb, gbg, ggb, ggg} Assuming that boys and girls are equally likely, find the probability of getting three children of all the same gender. 2 P three children of the same gender 0.25 8 Copyright © 2014, 2012, 2010 Pearson Education, Inc. Section 4.2-14 Simulations A simulation of a procedure is a process that behaves in the same ways as the procedure itself so that similar results are produced. Copyright © 2014, 2012, 2010 Pearson Education, Inc. Section 4.2-15 Probability Limits Always express a probability as a fraction or decimal number between 0 and 1. The probability of an impossible event is 0. The probability of an event that is certain to occur is 1. For any event A, the probability of A is between 0 and 1 inclusive. That is, 0 P( A) 1 . Copyright © 2014, 2012, 2010 Pearson Education, Inc. Section 4.2-16 Possible Values for Probabilities Copyright © 2014, 2012, 2010 Pearson Education, Inc. Section 4.2-17 Complementary Events The complement of event A, denoted by A, consists of all outcomes in which the event A does not occur. Copyright © 2014, 2012, 2010 Pearson Education, Inc. Section 4.2-18 Example 1010 United States adults were surveyed and 202 of them were smokers. It follows that: 202 P smoker 0.200 1010 202 P not a smoker 1 0.800 1010 Copyright © 2014, 2012, 2010 Pearson Education, Inc. Section 4.2-19 Rounding Off Probabilities When expressing the value of a probability, either give the exact fraction or decimal or round off final decimal results to three significant digits. (Suggestion: When a probability is not a simple fraction such as 2/3 or 5/9, express it as a decimal so that the number can be better understood.) All digits are significant except for the zeros that are included for proper placement of the decimal point. Copyright © 2014, 2012, 2010 Pearson Education, Inc. Section 4.2-20 Definition An event is unlikely if its probability is very small, such as 0.05 or less. An event has an usually low number of outcomes of a particular type or an unusually high number of those outcomes if that number is far from what we typically expect. Copyright © 2014, 2012, 2010 Pearson Education, Inc. Section 4.2-21 Odds The actual odds against event A occurring are the ratio P( A) / P( A), usually expressed in the form of a:b (or “a to b”), where a and b are integers having no common factors. The actual odds in favor of event A occurring are the ratio P( A) / P( A) , which is the reciprocal of the actual odds against the event. If the odds against A are a:b, then the odds in favor of A are b:a. The payoff odds against event A occurring are the ratio of the net profit (if you win) to the amount bet. payoff odds against event A = (net profit) : (amount bet) Copyright © 2014, 2012, 2010 Pearson Education, Inc. Section 4.2-22 Example If you bet $5 on the number 13 in roulette, your probability of winning is 1/38 and the payoff odds are given by the casino at 35:1. a. Find the actual odds against the outcome of 13. b. How much net profit would you make if you win by betting on 13? Copyright © 2014, 2012, 2010 Pearson Education, Inc. Section 4.2-23 Example - continued a. Find the actual odds against the outcome of 13. With P(13) = 1/38 and P(not 13) = 37/38, we get: P not 13 37 38 37 actual odds against 13 , or 37:1. 1 P 13 1 38 Copyright © 2014, 2012, 2010 Pearson Education, Inc. Section 4.2-24 Example - continued b. Because the payoff odds against 13 are 35:1, we have: $35 profit for each $1 bet. For a $5 bet, there is $175 net profit. The winning bettor would collect $175 plus the original $5 bet. Copyright © 2014, 2012, 2010 Pearson Education, Inc. Section 4.2-25