Question

Demand for your company’s product is influenced by two factors: economic conditions for the year (good, fair or poor) and the intensity of market competition (intense or moderate). Company analysts estimate a likelihood of 30% that economic conditions will be good, a 50% likelihood that they will be fair, and a 20% likelihood they will be poor. It is estimated that if economic conditions are good, there is an 80% probability that market competition will be intense (and so a 20% chance it will be only moderate). If economic conditions are only fair, there is a 50% chance that competition will be intense (and a 50% chance it will be moderate). Finally, if economic conditions are poor, there is only a 10% chance that competition will be intense (and a 90% chance it will be moderate). The company has a basic demand level of 5000 units, but its analysts estimate that good economic conditions will add 2000 units to this total, while fair economic conditions will add 1500. (Poor conditions will add nothing to basic demand.) In addition, moderate competition can be expected to add an extra 1000 units, while intense competition will subtract 1000 units. Using company demand as your random variable and X to represent values of the random variable:

a. list all possible X values.

b. show the full probability distribution.

c. compute the expected demand probability. (Hint: You might try a probability tree here.)

Answer 1

Tree diagram

Case 1: Economic Condition Good ;  Market competition intense

X = Basic Demand + Economic Condition Good + Market competition intense = 5000+2000-1000=6000

Probability of Economic Condition Good x Probability of Market competition intense given Economic Condition Good

= 0.3 x 0.8 = 0.24

P(X=6000) = 0.24

Case 2: Economic Condition Good ;  Market competition moderate

X = Basic Demand + Economic Condition Good + Market competition moderate = 5000+2000+1000=8000

Probability of Economic Condition Good x Probability of Market competition moderate given Economic Condition Good

= 0.3 x 0.2 = 0.06

P(X=8000) = 0.06

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Case 3: Economic Condition Fair ;  Market competition intense

X = Basic Demand + Economic Condition fair + Market competition intense = 5000+1500-1000=5500

Probability of Economic Condition fair x Probability of Market competition intense given Economic Condition fair  

= 0.5 x 0.5= 0.25

P(X=5500) = 0.25

Case 4: Economic Condition fair ;  Market competition moderate

X = Basic Demand + Economic Condition fair + Market competition moderate = 5000+1500+1000=7500

Probability of Economic Condition fair x Probability of Market competition moderate given Economic Condition fair

= 0.5 x 0.5 = 0.25

P(X=7500) = 0.25

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Case 5: Economic Condition poor ;  Market competition intense

X = Basic Demand + Economic Condition poor + Market competition intense = 5000+0-1000=4000

Probability of Economic Condition poor x Probability of Market competition intense given Economic Condition poor

= 0.2 x 0.1= 0.02

P(X=4000) = 0.02

Case 6: Economic Condition poor ;  Market competition moderate

X = Basic Demand + Economic Condition poor + Market competition moderate = 5000+0+1000=6000

Probability of Economic Condition fair x Probability of Market competition moderate given Economic Condition poor = 0.2 x 0.9 = 0.18

P(X=6000) = 0.18

a)

From the above,

List all the possible values of X from the above: 6000,8000,5500,7500,4000,6000

i.e

Possible values of X = 4000,5500,6000, 7500, 8000

b. Probability distribution

X	P(X)
4000	0.02
5500	0.25
6000	0.42
7500	0.25
8000	0.06

c. Expected demand = E(X)

Expected demand = 6330