Conditional Probability Problem, Probability of Pulling A Club after Red Card I am currently learning about conditional probability from "The Book of R" by Tilman Davies and I'm having trouble understanding a conditional probability problem. The problem is as follows:
You randomly draw a card, and after replacing it, you draw another. Let A be the event that the card is a club; let B be the event that the card is red. What is Pr(A|B)? That is, what is the probability the second card is a club, given the first one was a red card? Are the two events independent?
This assumes your standard 52 card deck. I understand that Pr(B) = 26/52, however where I get stuck is when the question asks about clubs which are black and not red. This means that it is impossible to pull a club out of the new sample space (now 26) because clubs are black and not red. The answer key to the book says the final value is 13/52 but I have absolutely no idea how that conclusion was reached. I have always struggled with understanding probability so any help anyone could provide I would greatly appreciate. 
 A: Wow, that's poor notation$^\dagger$ they have in the question; it doesn't distinguish which draw the events apply to properly.
Before you draw any cards you have 26 red cards, 13 clubs (and 13 spades). After the first draw is red, they say you replace it (presumably shuffling again), so you still have 26 red cards, 13 clubs and 13 spades. At the second draw there are 52 cards, of which 13 are clubs; the chance of drawing a club is $\frac{13}{52}$. (Since the first-drawn card was replaced it changes nothing; you're exactly in the situation you started with)
$\dagger$ A better notation would be: Let $R_i$ be the event "a red card was drawn at the $i$th card draw" and let $C_i$ be the event "a club was drawn at the $i$th card draw; then the problem is "What is $P(C_2|R_1)$?" which is useful to distinguish from various other possible questions. Ambiguous notation is a recipe for confusion.

In reply to comment:
The actual sample space for two draws is the $52\times 52$-element set consisting of all possible pairs of cards from $(A♡,A♡)$ to $(K♠,K♠)$. When you draw a red card on the first draw, you restrict attention to the $26\times 52$ set that has only red cards on the first draw. 
Given that, the events with a club on the second draw are the $26 \times 13$ cases that have a red on the first draw and a club on the second draw, and the conditional probability is $\frac{26\times 13}{26\times 52}$. Note that the first draw simply cancels out of numerator and denominator, because the two draws are independent; you can simply ignore it, because it contributes no information. 
[Notation that ignores that there are two draws encourages you to make the error of thinking the sample space only has 52 elements.]
A: 
You randomly draw a card, and after replacing it, you draw another.
  Let A be the event that the card is a club; let B be the event that
  the card is red. What is Pr(A|B)? That is, what is the probability the
  second card is a club, given the first one was a red card? Are the two
  events independent?

Let's interpret this paragraph for you. I believe the problem you are having is one of English. Specifically, the word given is not quite what you think it is. You ought to interpret the word given to only mean that if I tell you that A has happened, now what are the chances that B will happen.


*

*You have a deck of 52 cards

*You pull one out, it is a red card, you put it back in. (the text specifically says, "and after replacing it")

*You pull out a new card (from the same 52 card deck)

*What are the odds this new card is a club?


In some cases, the first activity might limit the sample size for the second event. In this particular case, it does not.
