I've made a spreadsheet to match your table (which doesn't take into account rounding to a whole number of people each time, but is close to the numbers we got before), and modified it to list probabilities of being selected exactly 0, 1, 2, 3, 4 times, or at least 1, 2, 3, 4 times. Of more interest, perhaps, at the moment, are the expected numbers for something like your specific conditions. Here are my results:

View attachment 12567
This is pretty close to the numbers you reported, so it actually does look quite random. (The numbers shown are the 250 times the probability of each event.)

Here is the formula in cell B6: =B$1*COMBIN(B$3,$A6)*(B$2/B$3)^$A6*(1-B$2/B$3)^(B$3-$A6)

This could also be done using the Binomial Distribution function: =B$1*BINOM.DIST($A6,B$3,B$2/B$3,0). But you like the math, ... .

As for changes: if you decrease the probability of being selected more than once, you will not be doing random selection. Indeed, if you eliminated the possibility of being selected more than once, you would be violating the point of the testing, as those people would know they are "safe".