Deriving the Probability That at Least One Person is COVID-19 Positive in A Group of Random People That Meet Together for Three Days

Solution 1:

I disagree with your analysis. As I see it, the critical factor is the total number of (random) people involved.

As you describe it, since $2$ of the people are replaced, there is an exposure of $(12)$ people.

Based on current incidence rates for my area,the probability of 1 person being COVID-19 positive in a group of 10 people is 19%.

Let $p = $ the probability that a specific person is COVID-19 positive.

Let $q = 1 - p$.

Then $0.81 = q^{(10)} \implies q \approx 0.979 \implies p \approx (1 - 0.979) = 0.021.$

Then, the probability of at least one person being COVID-19 positive, in a group of $12$ is

$1 - q^{12} \approx 0.223$.

The problem with your approach is that exposure on each of the $3$ days are not independent events, since you do not have $10$ different people, on each of the $3$ days.

Admittedly, I am making the assumption that if someone is not COVID-19 positive on day 1, they will also not be COVID-19 positive on day 3.

As a more graphic illustration:

which is more dangerous:

• taking a bus ride with the same $10$ people, $3$ days in a row:

• taking a bus ride, 3 days in a row, where there are $10$ different people on each of the $3$ days.