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Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 4-1 Business Statistics: A Decision-Making Approach 7 th Edition Chapter 4 Using Probability and Probability Distributions

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Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 4-2 Important Terms Random Variable - Represents a possible numerical value from a random event and it can vary from trial to trial Probability – the chance that an uncertain event will occur (always between 0 and 1) Experiment – a process that produces outcomes for uncertain events Sample Space (or event) – the collection of all possible experimental outcomes

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Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 4-3 Basic Rule of Probability Individual ValuesSum of All Values 0 ≤ P(E i ) ≤ 1 For any event E i where: k = Number of individual outcomes in the sample space e i = i th individual outcome The value of a probability is between 0 and 1 0 = no chance of occurring 1 = 100% change of occurring The sum of the probabilities of all the outcomes in a sample space must = 1 or 100%

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Simple probability The probability of an event is the number of favorable outcomes divided by the total number of possible outcomes. This assumes the outcomes are all equally weighted. Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 4-4

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Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 4-5 Visualizing Events Contingency Tables Tree Diagrams Red 2 24 26 Black 2 24 26 Total 4 48 52 Ace Not Ace Total Full Deck of 52 Cards Red Card Black Card Not an Ace Ace Not an Ace Sample Space 2 24 2 24

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Simple probability What is the probability that a card drawn at random from a deck of cards will be an ace? 52 cards in the deck 4 are aces The probability is 4/52 Each card represents a possible outcome 52 possible outcomes. Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 4-6

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Simple probability Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 4-7 There are 36 possible outcomes when a pair of dice is thrown.

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Simple probability Calculate the probability that the sum of the two dice will equal 5? Four of the outcomes have a total of 5: find out from the table (1,4; 2,3; 3,2; 4,1) Probability of the two dice adding up to 5 is 4/36 = 1/9 since there are 36 possibilities. Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 4-8

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Simple probability Calculate the probability that the sum of the two dice will equal 12? Only one (6,6) 1/36 Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 4-9

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Notation of Probability The probability of event A is denoted by P(A) Example: Suppose a coin is flipped 3 times. What is the probability of getting two tails and one head? The sample space consists of 8 sample points. S = {TTT, TTH, THT, THH, HTT, HTH, HHT, HHH} the probability of getting any particular sample point is 1/8. getting two tails and one head: A = {TTH, THT, HTT} P(A) = 1/8 + 1/8 + 1/8 = 3/8 Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 4-10

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Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 4-11 Independent and Dependent Events Independent: Occurrence of one does not influence the probability of occurrence of the other Dependent: Occurrence of one affects the probability of the other Probability Concepts

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Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 4-12 Independent Events A = heads on one flip of fair coin B = heads on second flip of same coin Result of second flip does not depend on the result of the first flip. Dependent Events X = rain forecasted on the news Y = take umbrella to work Probability of the second event is affected by the occurrence of the first event Independent vs. Dependent Events

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Rule of Multiplication The rule of multiplication applies to the situation when we want to know the probability that two events (Event A and Event B) both occur. Independent: P(A ∩ B) = P(A) P(B) Dependent: P(A ∩ B) = P(A) P(B|A) Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 4-13

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Independent Events If A and B are independent, then the probability that events A and B both occur is: p(A and B) = p(A) x p(B) What is the probability that a coin will come up with heads twice in a row? Two events must occur: a head on the first toss and a head on the second toss. The probability of each event is 1/2 the probability of both events is: 1/2 x 1/2 = 1/4. Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 4-14

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Independent Events What is the probability that the first card is the ace (put it back: replace) of clubs and the second card is a club (any club)? The probability of the first event is 1/52. The probability of the second event 13/52 = 1/4 (composed of clubs) 1/52 x 1/4 = 1/208 Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 4-15

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Dependent Events If A and B are not independent, then the probability of A and B both occur is: p(A and B) = p(A) x p(B|A) where p(B|A) is the conditional probability of B given A. Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 4-16

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What is Conditional Probability? A conditional probability is the probability of an event given that another event has occurred. Conditional probability for any two events A, B: Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 4-17 Notation

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Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 4-18 Conditional Probability Example No CDCDTotal AC.2.5.7 No AC.2.1.3 Total.4.6 1.0 What is the probability that a car has a CD player, given that it has AC ? we want to find P(CD | AC)

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Conditional Probability Example What is the probability that the total of two dice will be greater than 8 given that the first die is a 6? This can be computed by considering only outcomes for which the first die is a 6. Then, determine the proportion of these outcomes that total more than 8. Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 4-19 (continued)

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Conditional Probability Example There are 6 outcomes for which the first die is a 6. And of these, there are four that total more than 8. (6,3; 6,4; 6,5; 6,6) Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 4-20 (continued)

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Conditional Probability Example 6 outcomes for which the first die is a 6 There are four that total more than 8 The probability of a total greater than 8 given that the first die is 6 is 4/6 = 2/3. More formally, this probability can be written as: p(total>8 | Die 1 = 6) = 2/3(6,3; 6,4; 6,5; 6,6). Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 4-21 (continued)

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Conditional Probability Example What is the probability that the first card is the ace (not put it back: no replace) and the second card is also an ace? First: p(A) = 4/52 = 1/13. Of the 51 remaining cards, 3 are aces. So, p(B|A) = 3/51 = 1/17 Probability of A and B is: 1/13 x 1/17 = 1/221. Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 4-22 (continued)

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Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 4-23 Probability Concepts Mutually Exclusive Events If A occurs, then B cannot occur A and B have no common elements Black Cards Red Cards A card cannot be Black and Red at the same time. A B

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Rule of Addition We have two events, and we want to know the probability that either event occurs. Mutually exclusive: P(A or B) = P(A) + P(B) Not Mutually exclusive: P(A or B) = P(A)+ P(B) – P(A and B) Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 4-24

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Mutually Exclusive Events If events A and B are mutually exclusive, then the probability of A or B is p(A or B) = p(A) + p(B) What is the probability of rolling a die and getting either a 1 or a 6? impossible to get both a 1 and a 6 (that is, mutually exclusive). p(1 or 6) = p(1) + p(6) = 1/6 + 1/6 = 1/3 Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 4-25

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Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 4-26 A bag of candy contains 4 different flavors: lemon, strawberry, orange, and blueberry What is the probability that you will get a lemon OR an orange piece of candy? E 1 = lemon = (8/35) = 0.23 E 2 = orange = (5/35) = 0.14 Mutually Exclusive Events (continued) FlavorCount Lemon8 Strawberry12 Orange5 Blueberry10 TOTAL 35 P(E 1 or E 2 ) = P(E 1 ) + P(E 2 ) = 0.23 + 0.14 = 0.37

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Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 4-27 Not Mutually Exclusive Events P(A or B) = P(A)+ P(B) – P(A and B) AB P(A or B) = P(A) + P(B) - P(A and B) Don’t count common elements twice! ■ If the events are not mutually exclusive, AB +=

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Not Mutually Exclusive Example What is the probability that a card will be either an ace or a spade? p(ace) = 4/52 and p(spade) = 13/52 The only way both (ace and a spade) can be drawn is to draw the ace of spades. There is only one ace of spades: p(ace and spade) = 1/52. The probability of an ace or a spade can be: p(ace)+p(spade)-p(ace and spade) = 4/52 + 13/52 - 1/52 = 16/52 = 4/13 Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 4-28

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Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 4-29 Complement Rule The complement of an event E is the collection of all possible elementary events not contained in event E. The complement of event E is represented by E. Complement Rule: E E Or,

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Probability Types Marginal probability Involves only a single random variable, the outcome of which is uncertain Joint probability Involves two or more random variables, in which the outcome of all is uncertain Conditional probability Involves two or more random variables, in which the outcome of at least one is known Example: download “Probability Type Example ” Excel file Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 4-30

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Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 4-31 Chapter Summary Discussed probability terminology Described approaches to determining probabilities Reviewed common rules of probability Addition Rules (Rules 1 – 5) Multiplication Rules (Rules 8, 9) Defined conditional probability (Rules 6, 7) Used Bayes’ Theorem for conditional probabilities

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