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?