4. John sells ice cream bars at the local park. He buys the bars for $1.20 per dozen and sells them for $0.15 each. If it is a rainy day at the park, he can sell nothing; if is overcast, he sells one dezen bars; and if is sunny, he sells two dozen. He can return unsold bars, but on;y gets $0.05 each for them. (He can only buy in dozens).
1. Assume that John is a pessimist, what should he do? Buy:
a) 2
b) 0
c) 1.5
d) 3
e) None of these
2. Assume that John is an optimist (gambler), what should he do? Buy:
a) 0
b) 1
c) 1.5
d) 2
e) 2.5
3. A one-legged, red-nosed tramp approaches John in the park and tells him he can predict the next day’s weather perfectly after he has drunk a shot of “Ripple” wine (cost of one shot: $0.50). If the tramp is telling the truth, should John engage in this transaction?
a) Yes
b) No
c) Not enough information to answer this question.
4. Assume that John is using minimax regret criteria, how many dozen should he buy?
a) 0
b) 1
c) 1.5
d) 2
e) 2.5
5. If the weather bureau’s forecast for the next day is 30 percent chance of rain, 20 percent chance of being sunny, how many dozen bars should he buy for tomorrow?
a) 0
b) 1
c) 2
d) 3
6. The value of perfect information is:
a) 4
b) 20
c) 24
d) 45
e) 54
1. If John is a pessimist, he should consider the worst-case scenario, which is selling nothing if it is a rainy day. Therefore, he should buy 0 dozen bars (option b).
2. If John is an optimist or a gambler, he should consider the best-case scenario, which is selling two dozen bars if it is a sunny day. Therefore, he should buy 2 dozen bars (option d).
3. Without enough information about the tramp's prediction accuracy, it is not advisable for John to engage in this transaction. Therefore, the answer is c) Not enough information to answer this question.
4. Using the minimax regret criteria, John should choose the option that minimizes his maximum possible regret. To calculate regret, we need to find the difference between the best outcome for each decision and the actual outcome, then take the maximum regret for each decision. The options are:
- Buying 0 dozen: Maximum regret = $0.00 (selling nothing)
- Buying 1 dozen: Maximum regret = $0.00 (selling 1 dozen)
- Buying 1.5 dozen: Maximum regret = $0.60 (selling 2 dozen - $0.60 saved from returning unsold bars)
- Buying 2 dozen: Maximum regret = $0.60 (selling 2 dozen - $0.60 saved from returning unsold bars)
- Buying 2.5 dozen: Maximum regret = $3.00 (selling 2 dozen - $1.20 saved from returning unsold bars)
Therefore, John should buy 0 dozen bars (option a).
5. Based on the weather bureau's forecast, John should consider the probability of each scenario. If there is a 30 percent chance of rain, a 20 percent chance of being sunny, and the remaining 50 percent chance of being overcast, he should calculate the expected value for each option:
- Buying 0 dozen: Expected value = $0.00 (0.30 x $0.00 + 0.50 x $0.00 + 0.20 x $0.00)
- Buying 1 dozen: Expected value = $0.15 (0.30 x $0.00 + 0.50 x $0.15 + 0.20 x $0.00)
- Buying 2 dozen: Expected value = $0.30 (0.30 x $0.00 + 0.50 x $0.30 + 0.20 x $0.00)
- Buying 3 dozen: Expected value = $0.45 (0.30 x $0.15 + 0.50 x $0.30 + 0.20 x $0.00)
Based on the expected values, John should buy 3 dozen bars (option d).
6. The value of perfect information is the difference between the expected value with perfect information and the expected value without perfect information. Assuming each dozen costs $1.20, the calculation would be:
Value of Perfect Information = (Expected value with perfect information - Expected value without perfect information) x Number of dozens
Without perfect information, John's expected value is $0.45 (from question 5). However, with perfect information, he can make the optimal decision based on the accurate weather forecast. If the accurate forecast tells him whether it will rain or be sunny, he can buy the precise number of dozens to maximize his profit.
Assuming the accurate forecast costs nothing, the calculation would be:
Value of Perfect Information = (0.30 x ($0.15 x 2) + 0.20 x ($0.15 x 1)) - $0.45) x Number of dozens
Value of Perfect Information = ($0.09 + $0.03 - $0.45) x Number of dozens
Value of Perfect Information = $0.12 x Number of dozens
Therefore, the Value of Perfect Information is $0.12 x 3 (assuming John buys 3 dozen bars), which equals $0.36. The answer is e) $0.36