# Does randomly selecting WITHOUT replacement imply independent events?

After a few days of researching and going through previous posts, I'm very much unclear on this topic still. My question is whether or not randomly taking samples without replacement implies that these samples are independent. In a related post (Independent or dependent events, drawing cards without replacement), removing two cards from a deck, the events are said to be independent.

However, in an unrelated workbook I'm working through, when introducing the concept of Hypergeometric probability, they would describe these events as dependent. The question states: "An urn contains five red balls, and seven blue balls. Four balls are randomly selected without replacement. Determine the probability that exactly one of them is red." along with "Choosing each ball affects the probability that the following ball will be a certain color, because the sample space has changed. Thus, the selection of each ball is not an independent event." $$P(x) = \frac{{7 \choose 3} * {5 \choose 1}}{12 \choose 4} = .354$$ Any help would be greatly appreciated!

It is exactly as that second workbook says: Given that what you pick and don't put back changes the distribution of what's left, the events are not independent

It's helpful to think of simple toy examples.

Consider a bag with two balls: one white and one black. Suppose we take out the balls one after the other without replacement, and consider the events $$W_1=\text{the first ball is white}$$and $$W_2=\text{the second ball is white}.$$

Intuitively, it's obvious these are not independent events because if you know $$W_1$$ happened then you know $$W_2$$ cannot happen since there's only one white ball. This is what your book means by "the sample space has changed": after you remove the first ball, you alter the contents of the bag.

The probability of $$W_1$$ actually determines the probability of $$W_2$$ since $$\mathrm P(W_1)+\mathrm P(W_2)=1$$.

• @Bram28 Thank you for your response! However In this related question: (math.stackexchange.com/questions/1174057/…), drawing 2 cards from a deck, the events are said to be independent. Isn't that virtually the same scenario as the one described above? in which the events are actually dependent? Apr 17 '20 at 21:58
• Dear @Jorge, no! It's not the same scenario at all! In the scenario I present it is crystal clear that the events affect (even determine!) each other. In the linked question, however, this is not so: suppose you know your first card was a heart. The second event is not asking whether the second card will be a heart. It is asking a different question - about face cards. Apr 17 '20 at 22:05
• Thank you so much! So it was subtly in the question phrasing. But that definitely makes sense now! Apr 17 '20 at 22:08