The Alpha Company is considering testing a new product regionally before launching it nationally due to uncertainty about its success. If they test market it, the results will determine whether they nationally launch it or abandon it. They estimate a 84% chance of a positive test result if it's a good product, and 64% chance of a negative result if it's a bad product. The document asks to compute a probability table and decision tree for this scenario.

1. A
The Alpha Company is trying to decide whether to market a new product. As in
many new-product marketing situations, there is considerable uncertainty about
whether the new product will eventually “catch on.” Alpha believes that it might be
wise to introduce the product in a regional test market before introducing it
nationally. Therefore, the company’s first decision is whether to conduct the test
market. If Alpha decides to conduct the test market, it must then wait for the results
of the test market. Based on these results, it can then decide whether to market the
product nationally or abandon the new product altogether. On the other hand, if the
original decision is not to conduct a test market, then the decision – whether to
market the product nationally or abandon the new product – can be made
immediately. Furthermore, Alpha has estimated that without a regional test, the
new product is equally likely to be successful or to fail nationally. Alpha has also
estimated that if the new product is a “good” product (i.e., it will be successful
nationally), then the probability that the regional test is positive is 0.84 (which
implies that the probability of a negative regional test is 0.16); if the new product is
a “bad” product (i.e., it will fail nationally), then the probability that the regional test
is negative is 0.64 (which implies that the probability of a positive regional test is
0.36).
“To test market or not to test market?”
(a) Compute the probability table for this problem. (A hint on notation: please use S
and F to denote the event of a national success and failure, respectively; use P
and N to denote the event of a regional positive result and negative result,
respectively.) What is the conditional probability of a national success given that
the test market result is positive? What is the conditional probability of a
national failure given that the test market result is negative?
(b) Construct a decision tree for this problem. Be sure to clearly mark each node
(whether it is a decision node or event node), the decision or event of each
branch, and the probability for each event branch.
B
There are two sealed envelopes, each containing an amount of money; the amount of
money is either Rs.5K, Rs.10K, Rs.20K, or Rs.40K and everybody knows this.
Furthermore, it is also known to all that one envelope contains exactly twice as much
money as the other. The two envelopes are shuffled, and we give one envelope to you
and one to your opponent. After the envelopes are opened (but the amounts inside
are kept private) you and your opponent are given the opportunity to switch. If you
both want to switch, we let you.

2. Find the optimal strategy for playing this game. A strategy identifies what you would
do under all situations. Thus for this example you need to decide between switch and
don't switch for any amount of money you are looking at.
Amount of Decision (you need to select one)
money in your
envelope
Rs. 5K Offer to switch, Do not offer to switch
Rs. 10 Offer to switch, Do not offer to switch
Rs. 20 Offer to switch, Do not offer to switch
Rs. 40 Offer to switch, Do not offer to switch