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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!