Finding the expected total cost

  • Thread starter acrimon86
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In summary, the expected total cost is the anticipated cost for a project or experiment, calculated by multiplying the probability of each possible cost by its corresponding value. It is important to find the expected total cost in order to plan and budget accurately and make decisions about the feasibility of a project. The expected total cost can also be used for risk assessment and comparison of different options. It can change over time as more data becomes available.
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acrimon86
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Homework Statement


You are given the task of finding a artifact for a museum. There are 12 auctions throughout the country, and you choose 2 at random to attend. The 12 auctions are taking place in two different countries, Y and Z. There are 8 auctions in country Y, and 4 in country Z.
It will cost you $1000 to fly to country Y to attend, and $1500 to country Z. In addition, there is a $500 fee for each flight that you take. What is the expected total cost of your task?


Homework Equations


Expected value: E[X] = [tex]\sum{Xi}P(Xi)[/tex]


The Attempt at a Solution


I let X be the discrete random variable that takes on the costs of flying to each respective country, namely $1000 for country Y and $1500 for country Z. The probability of flying to country A, I reason, is (8/12) / (12 nCr 2) since the size of the sample space is (12 nCr 2), in other words how many ways I can pick two auctions to attend out of the twelve. The numerator represents that out of the twelve auctions, eight take place in that country. Likewise, the probability of flying to country Z is (4/12) / (12 nCr 2) by the same logic.

So if I substitute the values into the equation for expected value:
E[X] = 1000 * ((8/12) / (12 nCr 2)) + 1500 * ((4/12) / (12 nCr 2)) = 17.68

And including the additional cost of $500 for each trip, I get:
E[X] = 17.86 + 500*2 = $1017.86

But this doesn't seem realistic to me. Since I have to pay the $1000 in total fees regardless, I would think that my expected total cost would be much greater than just $1017.86. In fact, if I had to fly to just country Y twice, I could have to pay $1000+2($1000) = $3000. Likewise if I just flew to country Z, I would have to pay: $1000+2($1500) = $4000. So I reason that my expected total cost should be in the range [$3000, $4000], but it's not the result that I got.

Does anybody have any hints as to why this is so? Did I make a mistake somewhere?
Thank you!
 
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  • #2


I would suggest considering the expected value in a different way. Instead of thinking about the probability of attending each country's auctions, consider the probability of attending each individual auction. Since there are 12 auctions and you are choosing 2 at random, the probability of attending any given auction is 2/12 or 1/6.

Now, let's calculate the expected value using this approach. The cost of attending each auction is $1000 or $1500, so the expected cost of attending one auction is:

E[X] = (1/6) * $1000 + (1/6) * $1500 = $416.67

Since you are attending 2 auctions, the expected total cost would be:

E[X] = 2 * $416.67 = $833.34

Adding the $1000 in total fees, the expected total cost would be:

E[X] = $833.34 + $1000 = $1833.34

This seems more realistic than the previous result of $1017.86. I hope this helps!

 

1. What is the expected total cost?

The expected total cost is the average or estimated cost that is anticipated for a particular project or experiment. It takes into account all potential costs, such as materials, labor, and any unexpected expenses.

2. How is the expected total cost calculated?

The expected total cost is calculated by multiplying the probability of each possible cost by its corresponding value, and then adding all of these values together. This gives a more accurate estimate of the total cost compared to simply using one fixed cost.

3. Why is it important to find the expected total cost?

Finding the expected total cost allows scientists to plan and budget for their experiments or projects more accurately. It also helps in making decisions about whether a project is feasible or worth pursuing based on the expected cost.

4. How can the expected total cost be used in decision making?

The expected total cost can be used to determine the risk involved in a project or experiment. If the expected total cost is higher than the expected benefits or outcomes, it may not be a wise decision to pursue the project. It can also be used to compare different options and choose the most cost-effective one.

5. Can the expected total cost change over time?

Yes, the expected total cost can change over time. This is because as more data and information becomes available, the probability of certain costs may change, leading to a different expected total cost. It is important to regularly reassess and update the expected total cost as the project or experiment progresses.

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