Assignment
Problem 1
a) Say you know a family has two children, and further that at least one of them is agirl. What is the probability that they have two girls?
b) Say you know a family has two children, and further that at least one of them is agirl born on Monday. What is the probability that they have two girls?Should the answer be the same as in (a)?
Verify your answers using YASAI. You can make the following assumptions:- probability for a girl to be born on Monday: 1/7,- total number of families with two children: 5,000,000.
Problem 2
You make a perishable, volatile chemical for which you charge $2.25 per liter. You have 75 regular customers for the chemical, each of whom has an independent 90% chance of placing an order on any given day. You also get an average of 30 orders per day from other, non-regular customers; assume the number of non-regular customers per day has a Poisson distribution.
Every order is for one 20-liter container.
You produce the chemical by a process that produces 600 liters of the chemical at a cost of $1300. Each day, you can run the process any whole number of times. Because it is so unstable, any chemical left unsold at the end of the day must be recycled, at a cost of $0.35 per liter.
What is the best number of times to run the process? Consider four possible policies of running the process 1, 2, 3 or 4 times.
Problem 3
The management of a hotel is considering renting a portable filtration unit to process the water to make it drinkable. There are three possible filtration units:
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Unit 1
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Unit 2
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Unit 3
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Capacity (Gallons)
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800
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900
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1,000
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Cost
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$ 800
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$ 1000
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$ 1,300
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They are also considering not renting a filtration unit. If they don't rent a unit, or if the unit's capacity turns out to be insufficient to meet the hotel guests' demand for water, the hotel will have to supply bottled water to every occupied room.
Providing bottled water will cost the hotel a flat fee of $450, plus $5 per occupied room.
The hotel has 250 rooms. At present, they have 200 room reservations they consider "firm", with a negligible chance of cancellation. They have another 35 reservations they consider "doubtful", each with an independent 65% chance of resulting in an occupied room. They also expect to get some "last minute" requests for rooms, which they estimate to be Poisson with a mean of 30. Occupied rooms consume random amounts of water independently from one another. Each occupied room consumes the amount of water that is distributed according to the normal distribution with an average of 4 gallons of water and a standard deviation of 3.1.
Find an optimal plan for the management and also the probability of not having enough filtered water to meet this optimal demand.
Problem 4
Modify problem 4 in handout 11 that you were working on in class: Assume now that if a service cannot be completed the second day, there is 90% chance it will be finished at the end of the next day, in which case your total charge is $2,100. In case if the service isn't finished in three days, the truck is returned in the condition "not fixed", and the company receives only $1,000. In addition to the previously asked questions, provide statistics on the total number of trucks remaining "not fixed" during the testing period.