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4.1 THE BASICS OF PROBABILITY Page 171

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(Page 171) Probability – the chance that a particular event will occur. The probability value will be from 0 to 1. A value of 0 means that the event will not occur. A probability of 1 means that the event will occur. Anything between 0 and 1 reflects the uncertainty of the event occurring.

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CHANCE Chance is how likely it is that something will happen. To state a chance, we use a percent. 0 ½ 1 Probability Chance 0% 50% 100% Certain not to happen Equally likely to happen or not to happen Certain to happen

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Example: Walking out in the road and you see the rain coming: The probability of the rain today is 1 since it is for sure that it will rain.

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Example: If an airplane has a top speed of 450 mph, and the distance between city A and city B is 900 miles: The probability that the plane will make the trip in 1.5 hours is 0. city A City B

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Page 171 Experiment – a process that produces a single outcome whose result cannot be predicted with certainty. Sample Space – is the collection of all outcomes that can result from a selection, decision, or experiment.

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Flipping of 2 coins Outcomes: Head and Tail Sample Space: 1 st flip – Head, 2 nd flip – Head (H, H) 1 st flip – Head, 2 nd flip – Tail (H, T) 1 st flip – Tail, 2 nd flip – Head (T, H) 1 st flip – Tail, 2 nd flip - Tail (T, T) The collection of possible experimental outcomes is called the sample space.

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Defining the Sample Space Example 4.1 Page 171 Best –Bath Systems is interested in analyzing the sales of its three main product lines. Walk-in shower Jacuzzi Standard tub Step 1. Define the Experiment The experiment is the sale. The variable of interest is the product sold.

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Step 2. Defines the outcomes for one trial of the experiment. e1 = walk-in shower e2 = jacuzzi e3 = standard tub Step 3. Define the sample space. For single sale: SS = (e1, e2, e3) For two sales: SS = (e1, e2, e3, e4, e5, e6, e7, e8, e9) Outcome Sale 1 Sale 2 e1 walk-inwalk-in e2 walk-injacuzzi e3 walk-instandard e4 jacuzzi walk-in e5 jacuzzijacuzzi e6 jacuzzistandard e7 standard walk-in e8 standard jacuzzi e9 standard standard

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Using a Tree Diagram to Define the Sample Space Example 4-2, Page 172 Lincoln Marketing Research was retained to interview three television viewers to determine whether they objected to having ads for hard liquor on TV. The analyst assigned to the project is interested in listing the sample space. Step 1. Define the experiment. The experiment involves selecting three television viewers and posting the question: Would you object to hard liquor advertisements on television? Step 2. Determine the outcome. Possible outcomes are:Yes and No

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Step 3. Define the sample Space. Outcome Viewer 1 Viewer 2 Viewer 3 e1 no no no e2 no no yes e3 no yes no e4 no yes yes e5 yes no no e6 yes no yes e7 yes yes no e8 yes yes yes

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Page 172 Tree Diagram A tree diagram is often a useful way to define the sample space for an experiment that helps ensure that no outcomes are omitted. Example 4 – 3 Using a Tree Diagram to Define the Sample Space Step 1. Define the experiment. Step 2. Define the outcomes for a single trial of the experiment. Step 3. Define the sample space for three trials using a tree diagram.

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No No Yes No Yes No Yes No Yes No Yes Yes No Yes Viewer 1 Viewer 2 Viewer 3

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Page 180 Exercises 4 -3 If two customers are asked to list their choice of ice cream flavor from among vanilla, chocolate, and strawberry, list the sample space showing the possible outcomes. Outcome Customer 1 customer 2 e1 V V e2 V C e3 V S e4 C V e5 C C e6 C S e7 S V e8 S C e9 S S

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Customer 1 Customer 2 V C Vanilla S V Chocolate C S Strawberry V C S

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Page 180 Homework Answer 4 - 4

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Page 173 Event of Interest Event – is a collection of experimental outcomes. Example 4-3 Defining an Event of Interest KPMG Accounting is interested in its audit status. Step 1. Define the Experiment. The experiment consists of two randomly chosen audits. Step 2. List the outcomes associated with one trial of the experiment. audit done early audit done on time Audit done late

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Step 3. Define the sample space. OutcomeAudit 1 Audit 2 e1 early early e2 early on time e3 early late e4 on time early e5 on time on time e6 on time late e7 late early e8 late on time e9 late late Step 4. Define the event of interest. The event of interest at least one audit is completed late is composed of all outcomes in which one or more audits are late. This event (E): E = (e3, e6, e7, e8, e9) There are five ways in which one or more audits are completed late.

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Page 174 Mutually Exclusive Events : Two events are mutually exclusive if the occurrence of one event precludes the occurrence of the other event. Example 4 – 4 Mutually Exclusive Events Tech-Works, Inc. does contract assembly work for Hewlett-Packard. Each item produced on the assembly line can be thought of as an experimental trial. The manager at this facility can analyze their process to determine whether the events of interest are mutually exclusive.

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Step 1. Define the experiment. The experiment is producing a part on an assembly line. Step 2. Define the outcomes for a single trial of the experiment. On each trial the outcome is either a good or a defective item. Step 3. Define the sample space. Product 1Product 2 e1 goodgood e2 gooddefective e3 defectivegood e4 defectivedefective

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Step 4. Determine whether the events are mutually exclusive. Let E1 = both products are good Let E2 = at least one defective E1 and E2 are mutually exclusive because the two events have no outcomes in common. Having two good items and at the same time having at least one defective item is not possible.

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Page 176 Methods of Assigning Probability 1. Classical Probability – the method of determining probability based on the ratio of the number of ways an outcome or event of interest can occur to the number of ways any outcome or event can occur when the individual outcomes are equally likely. P(Ei) = No. of ways Ei can occur Total number of possible outcomes

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Page 176 Example 4 – 5 Classical Probability Assessment Galaxy Furniture: Each customer making a purchase exceeding $100 will qualify to select an envelope that has coupons for percentage discounts inside. There were 500 coupons. 400 were for 10% discount, 50 were for 20% discount, 45 were for 30% discount, and 5 were for 50% discount. Customers were interested in determining the probability of getting a particular discount amount. Step 1. Define the experiment. (Envelope is selected from a large drum) Step 2. Determine whether the possible outcomes are equally likely. Equally likely Step 3. Determine the total number of outcomes. (500 envelopes) Step 4. Define the event of interest. (First customer will get a 20% discount) Step 5. Determine the number of outcomes associated with the event of interest. (50 coupons with a discount of 20%)

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Step 6. Compute the classical probability. P(Ei) = P(20% discount) = = = 0.10 After the first customer selects an envelope from the drum, then probability that the next customer will get a particular discount will change, because the values in the denominator and numerator will change. Number of ways 20% can occur Total number of possible outcomes 50 500 Number of ways Ei can occur Total number of possible outcomes

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Page 177 Many games of chance are based on classical probability assessment. Classical probability is difficult to apply to most business situations. Example: Starting a business: SS = (Succeed, Fail) Find the probability that the business will succeed: P(Succeed) = ½ Is this true? Many factors go into determining the success or failure of a business. The possible outcome is not equally likely.

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Page 177 2. Relative Frequency Assessment– the method that defines probability as the number of times an event occurs divided by the total number of times an experiment is performed in a large number of trials. P(Ei) = where: Ei = The event of interest N = Number of trials Relative Frequency Assessment approach is based on actual observations. Number of times Ei occurs N

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Page 178 Example 4 – 6 Relative Frequency Probability Assessment Starbucks coffee sells caffeinated and decaffeinated drinks. The manager is interested in determining the probability that a customer will select a caffeinated versus a decaf drink. Step 1. Define the experiment. A randomly chosen customer will select between decaf and caffeinated drink. Step 2. Define the event of interest. the manager is interested in the event E1 customer selects caffeinated. Step 3. Determine the number of occurrences. the manager has observed 2,250 sales of decaf and caffeinated drinks. N = 2,250. There were 1,570 sales for caffeinated drinks

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Step 4. For the event of interest, determine the number of occurrences. There were 1,570 sales for caffeinated drinks. P(Ei) = = 0.6978 Based on the past history, the chance that a customer will purchase a caffeinated drink is just under 70% or a probability of 0.70. No. of times Ei occur = 1,570 N = 2,250

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Page 179 3. Subjective Probability – the method that defines probability of an event as reflecting a decision maker’s state of mind regarding the chances that the particular event will occur. A subjective probability is a measure of a personal conviction that an outcome will occur. The probability represents a person’s belief that an event will occur.

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Exercises: Page 180 1. Answer 4 – 5 Page 180 2. Answer 4 – 7 Page 180 3. Answer 4 – 8 Page 180 4. Answer 4 – 9 Page 180

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