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June 5th, 2015, 08:00 AM   #1
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Difficult Probability Problem

Hi,

I am struggling with a problem and I desperately need some help.

"A real estate developer owns 5 flats, that are to be construted. Flat1, Flat2, Flat3, Flat4 and Flat5. The developer promises his best friend one of these flats. Moreover they agree that the friend chooses his flat after all flats have been constructed. The developer, however, wants to sell the other 4 flats before the end of the construction. The developer doesn't know,though, which flat his friend will pick but he guarantees him his first choice so he decides the following: He asks 4 potential buyers to give him two preferences regarding which flat they want to buy. E.g. "Potential buyer 1" will give him (Flat1, Flat2) or (Flat3, Flat5)... meaning that this buyer is indifferent between Flat1 and Flat2 (Flat3 and Flat 5) and either flat would satisfy his demand.
Now, what is the probability that, after the friend has chosen his flat, all the other potential buyers end up getting one of the flats, they wanted?"

(Accidentally, I've posted the question earlier in high school math)
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June 7th, 2015, 03:40 PM   #2
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There are 10 combinations for each of the four people choosing 2 flats and 5 permutations for the person who gets first choice. Therefore given enough time, you could write out 10 * 10 * 10 * 10 * 5 = 50,000 possibilities to solve it, but nobody wants to do that.

For starters if the same combination of flats is chosen by three or more of the four people, not everybody can get his or her choice. For any one combination of 2 flats, the probability of 3 or more of the 4 people wanting it is 37/10,000. There are 5 flats, so if the combinations were independent, the probability of no combination being wanted by 3 or more of the 4 people would be ((10,000 - 37)/10,000)^5 = about 0.9816. However, since each flat is in exactly 4 of the combinations, I don't know if the combinations are independent.

It won't calculate the probability, but I might do 100 random number simulations to estimate it.
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Last edited by EvanJ; June 7th, 2015 at 03:47 PM.
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June 9th, 2015, 06:12 AM   #3
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hey thanks for your answer. Some of those points were very helpful. I am not sure, though, how you got to 37/10000.

For me the probability that 3 or 4 potential buyer end up having the same preference would be (4!/3!) * (1/10)^3 * (9/10) + 10* (1/10)^4 = 46/10000.

And another issue is: Even if only 2 buyers had the same preferences it might be possible that things don't work out for the developer. Imagine: 2 potential buyers both want flat 1 and flat2 and the friend of the developer wants flat 2 also.

Maybe you have accounted for that and I just didn't understand your argument properly? Anyways thanks for your help so far and don't be shy to correct my mistakes That is a very messy problem.
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June 9th, 2015, 11:48 AM   #4
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Quote:
Originally Posted by lOWIq View Post
hey thanks for your answer. Some of those points were very helpful. I am not sure, though, how you got to 37/10000.

For me the probability that 3 or 4 potential buyer end up having the same preference would be (4!/3!) * (1/10)^3 * (9/10) + 10* (1/10)^4 = 46/10000.

And another issue is: Even if only 2 buyers had the same preferences it might be possible that things don't work out for the developer. Imagine: 2 potential buyers both want flat 1 and flat2 and the friend of the developer wants flat 2 also.

Maybe you have accounted for that and I just didn't understand your argument properly? Anyways thanks for your help so far and don't be shy to correct my mistakes That is a very messy problem.
The problem with your equation is that the 10 should be a 1 because there is only 1 way for 4 people to all want the same pair.

For the other issue, I know I didn't account for that.

My random number simulations resulted in a probability of 216/500 = 0.432. The denominator is 500 because I did 100 simulations for the 4 people and then looked at if each simulation was compatible with the 4 people and with the person who gets first choice of 5 flats. This is not the exact probability, which I don't know how to calculate.
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June 12th, 2015, 03:03 AM   #5
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Hey thanks for the ballpark figure! Gives a good sense of what to expect!!! Problem itself is probably only solvable the hard way.

Generally I would agree that there is only one way for four people to all want the same pair. However, it might be the case that all 4 want (Flat1, Flat2) or that all 4 want (Flat2, Flat3). So don't we ignore those events if we don't account for the 10 possible combinations?

Many thanks!
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June 13th, 2015, 08:43 AM   #6
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Quote:
Originally Posted by lOWIq View Post
Hey thanks for the ballpark figure! Gives a good sense of what to expect!!! Problem itself is probably only solvable the hard way.

Generally I would agree that there is only one way for four people to all want the same pair. However, it might be the case that all 4 want (Flat1, Flat2) or that all 4 want (Flat2, Flat3). So don't we ignore those events if we don't account for the 10 possible combinations?

Many thanks!
That sounds like it could be right, but if you apply that argument to exactly 3 people wanting the same pair, it would then be 360 instead of 36 because there are 10 different pairs that could be wanted by 3 people. Either way I think the greater number (exactly 3 people) should be 36 times the smaller number (4 people) because for any one pair there are 4 times as many ways for exactly 3 people to want it than for all 4 people to want it, and if 1 person doesn't want that pair there are 9 ways for him to want another pair (4 * 9 = 36).
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November 18th, 2016, 07:02 PM   #7
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I'm going to try to do this problem in a spreadsheet. I'm going to do a little at a time and I may not post an answer for weeks.
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December 30th, 2016, 07:25 AM   #8
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Hi
I need help to write code using mathmatica
Can help me ???
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