# Probability of defective of 1 item after picking 5 items out

I had a quiz and one of the question is : If I have 25 items among them 10 are defective. The question is I pick 5 out of 25 and test them, what is the chance of 3 of them are defective.

The correct answer is using hypergeometric distribution. However my question is if I assume all 5 items are tested at the same time ( that means independent from each other) is it possible that it will become binomial distribution ?

Another question is if I pick randomly 5 items out of 25 and randomly number them as item 1,2,3,4,5. What is the probability of defective for item 1? My TA said that because we pick 5 out of 25 without replacement then the probability of defective of one item in the batch of 5 is no longer $\frac{10}{25}$ so I should use hyper geometric distribution in this case. I am quite confused since picking give you no new information how can the probability of defective of one random item changes? Can someone help me to clarify?

Thank you

For your second question, if you pick $5$ items and number them, the probability the first is defective is $\frac{10}{25}$. The probability the second is defective is also $\frac{10}{25}$. And so on. Many people find this unintuitive at first.
Of course the conditional probability the second is defective, given the first is, is not $\frac{10}{25}$. When we do sampling without replacement, we lose independence.