Problem Set Week 02

### Before Class 1

Quiz on Page lectures
Refresher quiz on probability.

### Before Class 2

Q93. A new device at your favorite big box store costs \$200. It has a one year guarantee from the manufacturer. The cashier offers you a special deal on a three year replacement warranty (assume it's good, will be honored, etc.) — \$40. You estimate that the chances of the device failing during second and third years is 25% and that the price of replacement by then will be \$150. Should you buy the service plan? What are the parameters of this decision model?

Q95. You never know what the weather is going to be around here. Somedays you need a sweater and some days you need sunglasses. The smart person, they say, always brings both. But suppose there is a definite hassle involved in bringing either (e.g., you ride a bike and space is tight). Sketch a decision tree that takes into account a cost to bringing either and a cost to not having either when you need them and the possibility that on a given day you might need one, the other, or both. Use plausible numbers of your own choice.

Q96. A college enrolls two types of students. Full-pay students pay \$40K tuition and half-pay students pay \$20K. At present the school spends \$1 million per year to recruit 200 students about 75% of whom are half-pay and 25% full-pay. A consultant submits a proposal to shift resources around and use GIS to target recruitment at zip codes that are more likely to yield full-pay students. She says there is a 75% chance that the results will be a slightly smaller class (190) but one with 40% full-pay and 60% half-pay. Unfortunately there is also a risk things won't turn out so well. There's a 25% chance that enrollment will drop to 170 and only 30% will be full pay. Use a decision tree to advise the college as to its best course of action.

Q97. There's an idea in philosophy called "Pascal's Wager" that describes a way of thinking about the existence of god. It goes like this. I have a choice to believe or not believe. And there is a chance that god exists and a chance that there is no god. If I believe and there is a god, I have a chance at eternal salvation. If I don't believe but there is a god, I suffer eternal damnation. If I do believe and it turns out there is no god, I will feel a bit of a chump, but the atheists can feel smug if opposite is the case. Sketch this situation as a decision tree. Should you believe in god?

Q98. Sketch the decision tree for the following scenario. I want to buy a used car. The car I am looking at is being offered at \$4000. The seller says it is in good shape all around. I look it over and agree, but you never know for sure. Suppose there is a 10% chance that it is a total dog and that buying it will be a \$2000 mistake. I know a mechanic who will give it a very thorough inspection, for a price. If my assumptions are correct, what's the most I should pay my mechanic?

Q101. If the farmer plants early and the spring is warm, she can get a 20% increase in her harvest. But if she plants early and there's a late frost she can lose 50% of her harvest. Historically, these late frosts happen one year in four (25% of the time). Use a decision tree to determine how much she would be willing to invest in a perfect forecast.

### In Class

Q93. A new device at your favorite big box store costs \$200. It has a one year guarantee from the manufacturer. The cashier offers you a special deal on a three year replacement warranty (assume it's good, will be honored, etc.) — \$40. You estimate that the chances of the device failing during second and third years is 25% and that the price of replacement by then will be \$150. Should you buy the service plan? What are the parameters of this decision model?

Q95. You never know what the weather is going to be around here. Somedays you need a sweater and some days you need sunglasses. The smart person, they say, always brings both. But suppose there is a definite hassle involved in bringing either (e.g., you ride a bike and space is tight). Sketch a decision tree that takes into account a cost to bringing either and a cost to not having either when you need them and the possibility that on a given day you might need one, the other, or both. Use plausible numbers of your own choice.

Q96. A college enrolls two types of students. Full-pay students pay \$40K tuition and half-pay students pay \$20K. At present the school spends \$1 million per year to recruit 200 students about 75% of whom are half-pay and 25% full-pay. A consultant submits a proposal to shift resources around and use GIS to target recruitment at zip codes that are more likely to yield full-pay students. She says there is a 75% chance that the results will be a slightly smaller class (190) but one with 40% full-pay and 60% half-pay. Unfortunately there is also a risk things won't turn out so well. There's a 25% chance that enrollment will drop to 170 and only 30% will be full pay. Use a decision tree to advise the college as to its best course of action.

Q97. There's an idea in philosophy called "Pascal's Wager" that describes a way of thinking about the existence of god. It goes like this. I have a choice to believe or not believe. And there is a chance that god exists and a chance that there is no god. If I believe and there is a god, I have a chance at eternal salvation. If I don't believe but there is a god, I suffer eternal damnation. If I do believe and it turns out there is no god, I will feel a bit of a chump, but the atheists can feel smug if opposite is the case. Sketch this situation as a decision tree. Should you believe in god?

Q98. Sketch the decision tree for the following scenario. I want to buy a used car. The car I am looking at is being offered at \$4000. The seller says it is in good shape all around. I look it over and agree, but you never know for sure. Suppose there is a 10% chance that it is a total dog and that buying it will be a \$2000 mistake. I know a mechanic who will give it a very thorough inspection, for a price. If my assumptions are correct, what's the most I should pay my mechanic?

Q101. If the farmer plants early and the spring is warm, she can get a 20% increase in her harvest. But if she plants early and there's a late frost she can lose 50% of her harvest. Historically, these late frosts happen one year in four (25% of the time). Use a decision tree to determine how much she would be willing to invest in a perfect forecast.

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