There's a .25 chance there will be a small flood and a .01 chance there will be a big flood. There's a piece of construction equipment in the town and the owner is trying to decide how to protect it.
A. Remove the equipment. This costs $3,800.
B. Create a wall for it. This costs $2,000, but will not protect it from a big flood. The equipment will be destroyed and the cost will be $60,000.
C. No plan at all. Hope that a flood doesn't come. It will cost $60,000 if a big flood comes and $10,000 if a small flood comes.
What should he do?
A. Remove the equipment. This costs $3,800.
B. Create a wall for it. This costs $2,000, but will not protect it from a big flood. The equipment will be destroyed and the cost will be $60,000.
C. No plan at all. Hope that a flood doesn't come. It will cost $60,000 if a big flood comes and $10,000 if a small flood comes.
What should he do?
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Calculate the expected cost in each case.
A. $3,800.
B. He always spends $2,000, and there's an 0.01 chance that he will have to spend $60,000. Therefore the expected cost is $2,000 + 0.01 * $60,000 = $2,600.
C. The expected cost is .25 * $10,000 + .01 * $60,000 = $3,100.
Since the expected cost is cheapest with plan B, that's probably the answer you're supposed to give.
If we're being pedantic, the plan that minimizes the expected cost is not necessarily the right choice. For example, suppose he can't afford to spend $60,000 -- when he can't pay, he loses the job, loses his house, and his family is forced to fight stray dogs for scraps of food. In that case, it might be worthwhile to remove the equipment entirely to avoid that risk. (To put it another way, the problem seems to assume that he is "risk-neutral," whereas real people are "risk-averse" in many practical scenarios.)
A. $3,800.
B. He always spends $2,000, and there's an 0.01 chance that he will have to spend $60,000. Therefore the expected cost is $2,000 + 0.01 * $60,000 = $2,600.
C. The expected cost is .25 * $10,000 + .01 * $60,000 = $3,100.
Since the expected cost is cheapest with plan B, that's probably the answer you're supposed to give.
If we're being pedantic, the plan that minimizes the expected cost is not necessarily the right choice. For example, suppose he can't afford to spend $60,000 -- when he can't pay, he loses the job, loses his house, and his family is forced to fight stray dogs for scraps of food. In that case, it might be worthwhile to remove the equipment entirely to avoid that risk. (To put it another way, the problem seems to assume that he is "risk-neutral," whereas real people are "risk-averse" in many practical scenarios.)