[50 points] Martin’s Service Station is considering entering the snowplowing bus

Question

[50 points] Martin’s Service Station is considering entering the snowplowing business for the coming winter. Martin can purchase either a snowplow blade attachment for the station’s pickup truck or a new heavy-duty snowplow truck. Martin has analyzed the situation and believes that either alternative would be profitable if the snowfall is heavy. Smaller profits would result if the snowfall is moderate, and losses would result if the snowfall is light. The following table gives the corresponding payoffs and the probabilities of heavy, moderate, and light snowfall.

Winter Snowfall

Heavy

Moderate

Light

Buy Blade

$3,750

$1,250

–$1,250

Buy Truck

$7,250

$2,250

–$8,750

Probability

40%

30%

30%

Martin can either purchase the equipment now or wait until the end of September. It will cost an extra $250 for the equipment if he waits until September (therefore each of his payoffs will be $250 less than those listed above). However, since September weather is a predictor of winter snowfall, if he waits he will be able to incorporate his knowledge of September weather conditions (normal or cold) and winter snowfall (heavy, moderate, or light) as shown in the following table.

Winter Snowfall

September Weather

Heavy

Moderate

Light

Normal

28%

24%

28%

Unusually Cold

12%

6%

2%

Note that the values in the table are joint probabilities, i.e. the first value of 28% corresponds to the probability that September weather is normal and that winter snowfall is heavy.

Use the following notation:

N = normal Sept, C = cold Sept, H = heavy snowfall, M = moderate snowfall, L = light snowfall.

i).Create the decision tree and calculate the probabilities you will need. Fill them in on the decision tree.

Hint: You will need to find the following probabilities (using the formulas from the Topics) in order to begin: P(HN), P(MN), P(LN), P(HC), P(MC), P(LC), P(N), and P(C).

ii).Complete the decision tree by calculating the expected values for each path. Fill them in on the decision tree. (Note, you only need to hand in the completed decision tree from part iii, but show steps/work for both parts ii and iii. Make sure your complete tree diagram lists all nodes, probabilities, expected values, and terminal node payoffs.)

iii).Using the expected value approach, state completely Martin’s optimal strategy.

iv).What is the probability of each of the possible payoffs that Martin could see based on the strategy from part iv?

Please answer all the questions

Winter Snowfall

Heavy

Moderate

Light

Buy Blade

$3,750

$1,250

–$1,250

Buy Truck

$7,250

$2,250

–$8,750

Probability

40%

30%

30%

Explanation / Answer

The following table gives the joint and marginal probabilities for September weather (normal or cold) and winter snowfall (heavy, moderate, or light).
Snowfall Heavy Snowfall Moderate Snowfall Light sum:
Normal Sept 28% 24% 28% 80%
Cold Sept 12% 6% 2% 20%
sum 40% 30% 30% 100

If Martin waits until the end of September, then the cost of the blade will increase by $ 250.

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