Overinvolvement Ratios
Step 1. Define the subset events from the problem:
3.5 Bayes’ Theorem 135 events, and let A1 be some other event. We can find the probability of Ei, given A1, by using Bayes’ theorem:
P1Eiu A12 = P1A1u Ei2P1Ei2 P1A12
The denominator can be expressed in terms of the probabilities of A1, given the various Eis, by using the intersections and the multiplication rule:
P1A12 = P1A1> E12 + P1A1> E22 + g + P1A1> EK2
= P1A1u E12P1E12 + P1A1u E22P1E22 + g + P1A1u EK2P1EK2 These results can be combined to provide a second form of Bayes’ theorem.
Bayes’ Theorem (Alternative Statement)
Let E1, E2, . . . , EK be K mutually exclusive and collectively exhaustive events, and let A be some other event. The conditional probability of Ei, given A, can be expressed as Bayes’ theorem:
P1Eiu A12 = P1A1u Ei2P1Ei2 P1A12
P1Eiu A12 = P1A1u Ei2P1Ei2
P1A1u E12P1E12 + P1A1u E22P1E22 + g + P1A1u EK2P1EK2 (3.15) where
P1A12 = P1A1> E12 + P1A1> E22 + g + P1A1> EK2
= P1A1u E12P1E12 + P1A1u E22P1E22 + g + P1A1u EK2P1EK2
The advantage of this restatement of the theorem lies in the fact that the probabilities it involves are often precisely those that are directly available.
This process for solving conditional probability and>or Bayes’ problems is summa-rized in Example 3.24.
Example 3.24 Automobile Sales Incentive (Bayes’ Theorem)
A car dealership knows from past experience that 10% of the people who come into the showroom and talk to a salesperson will eventually purchase a car. To increase the chances of success, you propose to offer a free dinner with a salesperson for all people who agree to listen to a complete sales presentation. You know that some people will do anything for a free dinner, even if they do not intend to purchase a car. However, some people would rather not spend a dinner with a car salesperson. Thus, you wish to test the effectiveness of this sales promotion incentive. The project is conducted for 6 months, and 40% of the people who purchased cars had a free dinner. In addition, 10%
of the people who did not purchase cars had a free dinner.
The specific questions to be answered are the following:
a. Do people who accept the dinner have a higher probability of purchasing a new car?
b. What is the probability that a person who does not accept a free dinner will pur-chase a car?
Solution
136 Chapter 3 Elements of Chance: Probability Methods
We have presented a logical step-by-step or linear procedure for solving Bayes’ lems. This procedure works very well for persons experienced in solving this type of prob-lem. The procedure can also help you to organize Bayes’ problems. However, most real problem solving in new situations does not follow a step-by-step, or linear, procedure.
Thus, you are likely to move back to previous steps and revise your initial definitions. In some cases you may find it useful to write out Bayes’ theorem before you define the prob-abilities. The mathematical form defines the probabilities that must be obtained from the problem description. Alternatively, you may want to construct a two-way table, as we did in Example 3.23. As you are learning to solve these problems, use the structure, but learn to be creative and willing to go back to previous steps.
P1: The customer purchases a car.
P2: The customer does not purchase a car.
Step 2. Define the probabilities for the events defined in Step 1:
P1P12 = 0.10 P1D1u P12 = 0.40 P1D1u P22 = 0.10 Step 3. Compute the complements of the probabilities:
P1P22 = 0.90 P1D2u P12 = 0.60 P1D2u P22 = 0.90
Step 4. Apply Bayes’ theorem to compute the probability for the problem solution.
a. We know that the sales promotion plan has increased the probability of a car purchase if more than 10% of those that had dinner purchased a car. Specifically, we ask if
P1P1u D12 7 P1P12 = 0.10 Using Bayes’ theorem, we find that
P1P1u D12 = P1D1u P12P1P12
P1D1u P12P1P12 + P1D1u P22P1P22
= 0.40 * 0.10 0.40 * 0.10 + 0.10 * 0.90
= 0.308
Therefore, the probability of purchase is higher, given the dinner with the salesperson.
b. This question asks that we compute the probability of purchase, P1, given that the customer does not have dinner with the salesperson, D2. We again apply Bayes’ theorem to compute the following:
P1P1u D22 = P1D2u P12P1P12
P1D2u P12P1P12 + P1D2u P22P1P22
= 0.60 * 0.10 0.60 * 0.10 + 0.90 * 0.90
= 0.069
We see that those who refuse the dinner have a lower probability of purchase.
To provide additional evaluation of the sales program, we might also wish to compare the 6-month sales experience with that of other dealers and with previ-ous sales experience, given similar economic conditions.
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Example 3.25 Market Research (Bayes’ Throrem)
Blue Star United, a major electronics distributor, has hired Southwest Forecasters, a market research firm, to predict the level of demand for its new product that combines cell phone and complete Internet capabilities at a price substantially below its major competitors. As part of its deliverables, Southwest provides a rating of Poor, Fair, or Good, on the basis of its research. Prior to engaging Southwest Blue Star, management concluded the following probabilities for the market-demand levels:
P1Low2 = P1s12 = 0.1 P1Moderate2 = P1s22 = 0.5 P1High2 = P1s32 = 0.4 Southwest completes its study and concludes that the market potential for this product is poor. What conclusion should Blue Star reach based on the market-study results?
Solution A review of the market-research company’s records reveals the quality of its past predictions in this field. Table 3.12 shows, for each level of demand outcome, the proportion of Poor, Fair, and Good assessments that were made prior to introducing the product to the market.
Table 3.12 Proportion of Assessments Provided by a Market-Research Organization Prior to Various Levels of Market Demand (Conditional Probabilities)
Market Demand That Actually Occurred After Assessment Was Provided Assessment Low Demand (s1) Moderate Demand (s2) High Demand (s3)
Poor 0.6 0.3 0.1
Fair 0.2 0.4 0.2
Good 0.2 0.3 0.7
For example, on 10% of occasions that demand was high, the assessment prior to market introduction was Poor. Thus, in the notation of conditional probability, denot-ing Low, Moderate, and High demand levels by s1, s2, and s3, respectively, it follows that
P1Poor u s12 = 0.6 P1Poor u s22 = 0.3 P1Poor u s32 = 0.1 Given this new information, the prior probabilities
P1s12 = 0.1 P1s22 = 0.5 P1s32 = 0.4
for the three demand levels can be modified using Bayes’ theorem. For a low level of demand, the posterior probability is as follows:
P1s1u Poor2 = P1Poor u s12P(s12
P1Poor u s12P1s12 + P1Poor u s22P1s22 + P1Poor u s32P1s32
= 10.62 10.12
10.62 10.12 + 10.32 10.52 + 10.12 10.42 = 0.06 0.25 = 0.24
Similarly, for the other two demand levels, the posterior probabilities are as follows:
P1s2u Poor2 = 10.32 10.52
0.25 = 0.6 P1s3u Poor2 = 10.12 10.42 0.25 = 0.16
Based on this analysis we see that the probability for high demand is now reduced to 0.16, and the most likely outcome is moderate demand with a posterior probability of 0.6.
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