It all depends on how many rounds they test in. If it’s only one round, and 4 of 16 are chosen, then yes, 1/4. But let’s say they test every round, and it’s always the same chance, 1/4, each round. If you made it through 3 rounds, your chances of being tested once is 1/4 + 1/4 + 1/4, or 3/4, or 75% chance of being tested once.
Well no; those probabilities don't add (otherwise after 4 rounds you'd be at 100%, clearly wrong.)
At 1/4 per round, the chance of getting tested exactly once out of 3 rounds is 1/4 x 3/4 x 3/4 x 3 = 42%
I.e., 25% of getting tested in any 1 round, but not the other 2.
Or, from the other direction, the probability of not getting tested at all is .75 x .75 * .75 = .42
So the chances of getting tested *at least* once is about 58%
For exactly once, subtract:
prob. of getting tested every round: .25 ** 3 = 2%
prob. of getting tested in rounds 1 + 2 = .25 * .25 * .75 = 5%
prob. of getting tested in rounds 1 + 3 = .25 * .75 * .25 = 5%
prob. of getting tested in rounds 2 + 3 = .75 * .25 * .25 = 5%
Which comes out to the same 42% ( 58 - 2 - 5 - 5 - 5 = 41 because of rounding errors.)
I think.