Probability question involving independent events and expected value

Question

I am working on the following problem:

A company sends products in boxes of $12$ units each. Before shipping, $3$ random units from each box are tested. If any of the $3$ are defective, the entire box is held back. Each unit has an independent probability of $0.2$ of being defective. What percentage of shipments are held back?

My thinking is that regardless of the size of the shipment, the chance of at least one unit being defective is $1-0.8^3 = 0.488$ ($1$ minus the probability of all $3$ units passing the test), this would lead me to believe that the answer is $48.8\%$.

Is this solution correct? I'm a bit uncertain since my answer is independent of the shipment size, but I can't find any flaw in my reasoning?

Answer 1

As others have pointed out, you are correct. The probability does not depend on the size of the box.

If you are unsure about the intuition, consider the following equivalent problem :

Take three coins and toss them. What is the probability of getting 3 "Heads"? (of course, it is $1/8$). Now, what if the coins were first drawn from a box containing 12 coins? from a box containing 1000 coins?

Answer: always $1/8$, regardless of the size of the box!