Amy Goodburn is among a handful of Huskers who are beating the odds this semester.
Through the first three weeks of the University of Nebraska–Lincoln’s random COVID-19 testing, Goodburn is one of 239 students, instructors and staff who have been selected thrice. Goodburn’s “luck” is the talk of Zoom meetings and has the senior associate vice chancellor and dean of undergraduate education pondering other odds-shattering opportunities.
“I tweeted yesterday that I’m going to have to buy a lottery ticket,” said Goodburn, who co-led the committees that shaped the university’s COVID-19 response for the fall and spring semesters. “It’s a little weird receiving the (need to test) emails that I helped draft. But, truthfully, I don’t mind.
“I’m happy to continue doing my part to support the health and safety of our Husker community.”
As it turns out, any statistician, mathematician or researcher would shrug at Goodburn’s three-time selection. They also wouldn’t be entirely shocked if it extended for another week — maybe two.
“The university’s testing program is using a basic random sample, with the same pools and equal probability of an individual being selected each week,” said Judy Walker, associate vice chancellor and professor of mathematics. “When you work out the math, we would expect a certain number of repeat selections through at least the first few weeks.”
The weekly random COVID-19 testing (which launched Feb. 18) selects 20% (one in five) from two pools — 20,000 students and 5,000 faculty/staff. That translates into weekly selections of 4,000 students and 1,000 faculty/staff. And, those 5,000 names are returned to their respective pools before the random selection is made in each subsequent week.
As the probability of weekly selection remains steady at 20%, factoring the odds shifts to a (relatively) simple exponent formula of 0.20^(number of weeks). Odds of being selected in all of the first four weeks would be:
- Weeks 1-2: 0.20 x 0.20 = 0.04, or 4%, or 1 in 25;
- Weeks 1-3: 0.20 x 0.20 x 0.20 = 0.008, or 0.8%, or 1 in 125; and
- Weeks 1-4: 0.20 x 0.20 x 0.20 x 0.20 = 0.0016, or 0.16%, or 1 in 625.
When compared to the pool sizes, those odds translate into repeat selections of:
- Weeks 1-2: 800 students and 200 faculty/staff;
- Weeks 1-3: 160 students and 40 faculty/staff; and
- Weeks 1-4: 32 students and 8 faculty/staff.
“We could also expect one or two faculty and staff, and six or seven students selected in all of the first five weeks,” Walker said. “And, there will likely be one student who is selected the first six weeks in a row. That Husker ought to earn a prize.”
The probability of being selected six consecutive weeks is .000064, or .0064%, or 1 in 15,625.
Overall, the probability that Huskers will be selected multiple times (but not consecutively) throughout the spring semester are high and should be expected. Conversely, the chance of not being selected at all during 11 weeks of randomized testing this semester is about 11%.
“We would expect about 537 employees and about 2,147 students not to be selected at all by the end of the semester,” Walker said.
And, as for Goodburn (and the 238 other Huskers selected in the first three weeks), her odds of getting tested every week through the end of the semester are 1 in 50 million.