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# The math of school lotteriesPrev TopicNext Topic

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• United States
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This topic might be familiar to some depending on your kids' ages and the school options in your area.

Scenario A: Suppose a magnet school or special program has 100 spots open and students will be selected by a lottery. Say there are 2500 kids who are eligible for the lottery. With simple math, it's easy to see that each kid's chance of getting selected is 100/2500 = 4%.

Scenario B: There are still 100 spots open and 2500 eligible kids, but with a hitch. The school is partly supported by the local University of State. The magnet school sets aside 10 of the 100 spots for eligible kids who are children of UoS faculty. Of the 2500 eligible students, 25 are children of faculty. The magnet school first holds a lottery for the  10 faculty children spots, and then a lottery for the 90 remaining spots. The 15 faculty children who are not selected in the first lottery stay in the pool of 2490 kids for the second general lottery drawing.

In this scenario, a child of a faculty member has a 35/83 = 42.17% chance of getting a spot. (The math is 10/25 + (15/25)x(90/2490) = 35/83.)

A kid who is not a child of a faculty member has a 90/2490 = 3/83 = 3.61% chance of getting selected.

The first question is: In Scenario B, is it possible hold just one lottery, but weight the lottery entries of children of faculty so that they have a 35/83 chance of getting selected in the single lottery?

The idea is to find the number K, such that

K x 100/((K x 25) + 2475) = 35/83

which has the solution K = 35/3 ≈ 11.67. In other words, every child of a faculty member gets 35/3 entries into the lottery. Under this scheme, it still works out that kids who don't have a faculty parent have only a 3/83 chance of getting selected.

The second question is: Under Scenario B with the two separate lotteries, the expected number of faculty children among the 100 spots is 875/83, or about 10.54. What is the expected number of faculty children among the 100 spots when you hold just a single lottery but give faculty children a weight of 35/3?

• Kentucky
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Hopefully after this question is answered someone can explain the NBA draft lottery.

"When will we ever learn?"

• United States
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Quote: Originally posted by Stack47 on Nov 18, 2022

Hopefully after this question is answered someone can explain the NBA draft lottery.

Yeah, and how to solve the problem of intentionally tanking.

• United States
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December 14, 2019
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Quote: Originally posted by cottoneyedjoe on Nov 17, 2022

This topic might be familiar to some depending on your kids' ages and the school options in your area.

Scenario A: Suppose a magnet school or special program has 100 spots open and students will be selected by a lottery. Say there are 2500 kids who are eligible for the lottery. With simple math, it's easy to see that each kid's chance of getting selected is 100/2500 = 4%.

Scenario B: There are still 100 spots open and 2500 eligible kids, but with a hitch. The school is partly supported by the local University of State. The magnet school sets aside 10 of the 100 spots for eligible kids who are children of UoS faculty. Of the 2500 eligible students, 25 are children of faculty. The magnet school first holds a lottery for the  10 faculty children spots, and then a lottery for the 90 remaining spots. The 15 faculty children who are not selected in the first lottery stay in the pool of 2490 kids for the second general lottery drawing.

In this scenario, a child of a faculty member has a 35/83 = 42.17% chance of getting a spot. (The math is 10/25 + (15/25)x(90/2490) = 35/83.)

A kid who is not a child of a faculty member has a 90/2490 = 3/83 = 3.61% chance of getting selected.

The first question is: In Scenario B, is it possible hold just one lottery, but weight the lottery entries of children of faculty so that they have a 35/83 chance of getting selected in the single lottery?

The idea is to find the number K, such that

K x 100/((K x 25) + 2475) = 35/83

which has the solution K = 35/3 ≈ 11.67. In other words, every child of a faculty member gets 35/3 entries into the lottery. Under this scheme, it still works out that kids who don't have a faculty parent have only a 3/83 chance of getting selected.

The second question is: Under Scenario B with the two separate lotteries, the expected number of faculty children among the 100 spots is 875/83, or about 10.54. What is the expected number of faculty children among the 100 spots when you hold just a single lottery but give faculty children a weight of 35/3?

Man I'm glad I don't have young kids anymore. Remember how simple it was in the old days when you either sent your kids to the closest neighborhood public school or the Catholic school?

I ran simulations of your weighted drawing and the average number of faculty kids was about 8.89. I didnt expext it to be more than 1 kid fewer, so that is interesting. The average must be lower because the single drawing set up has the possibility of no faculty kids being selected.

If you're faculty, you obviously prefer the double drawing system. But I think school admin would also prefer the double drawing because weights are a pain in the but to deal with, and you get too much variance in the number of faculty kids with the single drawing set up. Dont fix what ain't broke.

• United States
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Quote: Originally posted by db101 on Nov 20, 2022

Man I'm glad I don't have young kids anymore. Remember how simple it was in the old days when you either sent your kids to the closest neighborhood public school or the Catholic school?

I ran simulations of your weighted drawing and the average number of faculty kids was about 8.89. I didnt expext it to be more than 1 kid fewer, so that is interesting. The average must be lower because the single drawing set up has the possibility of no faculty kids being selected.

If you're faculty, you obviously prefer the double drawing system. But I think school admin would also prefer the double drawing because weights are a pain in the but to deal with, and you get too much variance in the number of faculty kids with the single drawing set up. Dont fix what ain't broke.

You're probably right about two simple drawings being easier to administer than one drawing with complicated weights. I got the same expected value when I took random samples, and it wasn't intuitive at first how to even simulate a drawing with non-integer weights on some entries. The key is to take out all of an applicant's entries if one of their entries is picked because you can't have the same applicant picked twice. Not sure how you set yours up, but what I did was pick a random number over an interval of length 'X' and then if the number was less than 'Y' it counted as a non-faculty applicant getting selected and if was at least 'Y' it counted as a faculty applicant. Then 'X' and 'Y' were adjusted accordingly to reflect either 1 or 35/3 entries being eliminated from the pool. Rinse and repeat 99 more times.

If you add more special categories for reserved slots, then running a series of drawings becomes easier than a single drawing with lots of different weight categories. For example, if their were two special categories A and B, you'd need 4 classes of weights: one for people in neither category, one for people in A but not B, one for people in B but not A, and one for people in both A and B.

• United States
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Quote: Originally posted by cottoneyedjoe on Nov 20, 2022

You're probably right about two simple drawings being easier to administer than one drawing with complicated weights. I got the same expected value when I took random samples, and it wasn't intuitive at first how to even simulate a drawing with non-integer weights on some entries. The key is to take out all of an applicant's entries if one of their entries is picked because you can't have the same applicant picked twice. Not sure how you set yours up, but what I did was pick a random number over an interval of length 'X' and then if the number was less than 'Y' it counted as a non-faculty applicant getting selected and if was at least 'Y' it counted as a faculty applicant. Then 'X' and 'Y' were adjusted accordingly to reflect either 1 or 35/3 entries being eliminated from the pool. Rinse and repeat 99 more times.

If you add more special categories for reserved slots, then running a series of drawings becomes easier than a single drawing with lots of different weight categories. For example, if their were two special categories A and B, you'd need 4 classes of weights: one for people in neither category, one for people in A but not B, one for people in B but not A, and one for people in both A and B.

Not sure I can support a public charter or public magnet school having so many reserved spots in a lottery and tbh I didnt even know it was a thing. Your faculty example is borderline iffy, but I can see the justification if the Uni partly funds the school. And children of professors are probably better students academically.

I'd draw the line at reserving spots for students with characteristics they cant control, like their race or how much money their parents make.

• United States
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Quote: Originally posted by db101 on Nov 21, 2022

Not sure I can support a public charter or public magnet school having so many reserved spots in a lottery and tbh I didnt even know it was a thing. Your faculty example is borderline iffy, but I can see the justification if the Uni partly funds the school. And children of professors are probably better students academically.

I'd draw the line at reserving spots for students with characteristics they cant control, like their race or how much money their parents make.

I don't disagree with you. It can be a touchy issue. These policies are usually set by local school boards, so it might give you something to think about next time you're deciding which ones to vote for.

• United States
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Actually I might have to take that back. I'm aware of sibling preference in admissions. Unfair, yeah, but justified. It would be worse to foster jealousy and resentment in the home. Can you imagine if someone had twins and one got a spot thru the lottery and the other didn't, and the parents sent one to the good school and the other to the crappy school? I think if you didn't manage to beg for a second spot the only option would be to send both to the crappy school.

• Kentucky
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Quote: Originally posted by cottoneyedjoe on Nov 20, 2022

Yeah, and how to solve the problem of intentionally tanking.

The Warriors started tanking early this season.

"When will we ever learn?"

• United States
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March 28, 2019
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Quote: Originally posted by db101 on Nov 21, 2022

Actually I might have to take that back. I'm aware of sibling preference in admissions. Unfair, yeah, but justified. It would be worse to foster jealousy and resentment in the home. Can you imagine if someone had twins and one got a spot thru the lottery and the other didn't, and the parents sent one to the good school and the other to the crappy school? I think if you didn't manage to beg for a second spot the only option would be to send both to the crappy school.

I agree with you about sibling preference being important, at least when children are elementary aged. I started thinking about this topic of school lotteries after a conversation with a friend whose grandchild was applying for a spot in a public magnet school  or charter school lottery. They were complaining that the school had set aside a fixed number of spots for applicants who lived in the neighborhood, and that it was unfair because people could game the system by renting a small apartment and using it as the mailing address. I wonder if anyone would really go that far. Maybe if there was a big difference in quality between the normal public schools and the special public schools.

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