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# Probability :Decision Tree

2. The WP Company (hereinafter, WPC) is trying to decide whether to market a new product. As in many new product situations, there is considerable uncertainty about whether the new product will eventually "catch on" (a technical marketing term meaning "sell a lot"). WPC believes it might be smart to introduce the product in a regional "test market" before introducing it nationally. The first decision is whether to conduct the test market or not.

If WPC does conduct the test market, it estimates the net cost will be \$100,000, regardless of test market results (it is almost all fixed costs). WPC will incur a one-time cost of \$7 million if they decide to go to national marketing, whether they do that right away (no test market) or wait until they see the results of test-marketing. WPC makes \$18 on each unit it sells.

WPC classifies the national market size as "great" (600K units sold) or "awful" (90K units sold). If there is no test market information available, WPC estimates the probabilities for these outcomes to be 0.70 and 0.30, respectively.

In addition, WPC has the following historical data available from previous test markets on similar products:

? Of products that eventually became "great" in the national market, 85% had "strong" test market results and 15% were "weak" in the test market.
? Of products that eventually became "awful" in the national market, these probabilities were 20% and 80% for strong and weak, respectively.

WPC wants to use a "decision tree" approach to find the best strategy and it will go with the strategy that yields the highest expected profit.

a) Suppose that WPC is only considering going directly to the national market or not (ignore the test market for right now). Which of these two alternatives has the highest expected return?

b) Suppose when considering these two options, someone offers WPC "perfect information" about the national market. How much should they be willing to pay for this information?

If WPC conducts the test market, they will need to "update" the probabilities of a great and awful market based on the test results using Bayes theorem. The "prior probability" for a great market is 70%; if the test market comes back "strong," what is the updated probability of a great market? Using Bayes theorem, we have Pr(great) = .7, the Pr (Strong | great ) = .85 and the Pr (Strong | awful ) = .2; we want to calculate the Pr ( great | strong ), which we can do from these probabilities. Likewise, we can calculate the Pr ( great | weak ) and the overall probabilies Pr (strong) and Pr (weak) for the test market results.

c) Using the approach discussed above, consider WPC's evaluation of the test market: they have to pay \$100,000 to conduct the test, wait for the results, decide whether to go forward with the national market based on the new updated probabilities of the performance of their product nationally (which we calculate using Bayes theorem). What is the expected return associated with going forward with the test market?

d) Comparing WPC's major alternatives of doing nothing, going forward with the national market with no test market, or going forward with the test market, which has the highest expected return?

#### Solution Summary

Word file contains Decision Tree and detailed computations of highest expected return.

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