**ME30B Exam 2001/2002
Quest 2.3 (i)**

This question is taken from the ME30B Exam 2001/2002 found here (anyone can access this at the UWI website)

__Solution:__

We first draw the decision tree by using decision tree notation - the square representing a decision and the circle representing an event (i.e. a probability is associated with it). We have assumed in this case that the company initially has $0 assets and thus if we did the marketing survey and the products were not marketed then the company would have a final asset position of - $200,000.

Please note I have done this very quickly - there maybe algebra/arithmetic mistakes, so double check and make sure, the solution methodology remains the same!

We have added the payoffs of both Product 1 and Product 2, since there were no difference in their probability of demand.

Remember the figures on the left represent the expected value of each outcome, for example, for the Test Market Branch, if the market is Favorable, and the product is Marketed, and the Demand is Low, the expected value of the Low outcome will be: Payoff of Product 1 + Payoff from Product 2 - Marketing Study Cost = $300,000 + $1,000,000 - $200,000 = $1,100,000.

We are given the prior probabilities for Low Demand (L), Medium Demand (M) and High Demand (H).

P(L) = 0.3, P(M) = 0.6 and P(H) = 0.1

We are also given the likelihoods (conditional probabilities) of the market research being favorable given the demand.

P(F|L) = 0.3 P(U|L) = 0.7

P(F|M) = 0.6 P(U|M) = 0.4

P(F|H) = 0.8 P(U|H) = 0.2

From the decision tree, the posterior probabilities/ conditional probabilities we need are (start from the right of the tree and move to your left – look at the decision at the first circle from your right and then the decision at the second circle):

P(L|F), P(M|F), P(H|F), P(L|U), P(M|U), P(H|U).

We will also need the P(F) and P(U).

If you recall from conditional probability P (A∩B) = P(B) × P(A|B)

Therefore to find these probabilities we first need to find the probabilities of the intersections.

P(F∩L) = P(L) × P(F|L) = (0.3) × (0.3) = 0.09

Similarly:

P(F∩M) = (0.6) × (0.6) = 0.36

P(F∩H) = (0.1) × (0.8) = 0.08

Therefore, **P(F) = P(F∩L)
+ P(F∩M) + P(F∩H) = 0.09 +0.36 + 0.08 = 0.53**

(The marginal probability is the sum of all the intersections of one particular outcome)

Now the

P(U∩L) = P(L) × P(U|L) = (0.3) × (0.7) = 0.21

Similarly

P(U∩M) = (0.6) × (0.4) = 0.24

P(U∩H) = (0.1) × (0.2) = 0.02

Therefore,

**P(U) = P(U∩L) + P(U∩M)
+ P(U∩H) = 0.21 + 0.24 + 0.02 = 0.47**

We can now find:

P(L|F), P(M|F), P(H|F), P(L|U), P(M|U), P(H|U).

P(L|F) = P(L∩F) / P (F) = 0.09/ 0.53 = 0.17

Similarly

P(M|F) = 0.36/0.53 = 0.68

P(H|F) = 0.08/0.53 = 0.15

P(L|U) = 0.21/0.47 = 0.45

P(M|U) = 0.24/0.47 = 0.51

P(H|U) = 0.02/0.47 = 0.04

Based on these probabilities we can fill in the information into our decision tree. The solution is: not to do a market study but market directly the products, and should receive a profit of $2,060,000.

Remember, to calculate the expected value of an event we multiply the probability of each of its outcome by the outcome's expected value, and sum these values e.g. for the Not Test Marker Branch, let's look at the circle for when the products are Marketed. The expected value is (0.3 × $1,300,000 + 0.6 × $2,100,000 + 0.1 × $4,100,000 = $2,060,000).

Remember, we always choose a decision (i.e. represented by the square), which has the higher value (in the case of maximizing profit). For example for the first square where we have Test Market or Not to Test Market, the value of Test Market is $1,855,320 and for Not to Test Market is $2,060,000. We choose the higher between the two, in this case is $2,060,000 - Not to Test Market.