2 decks of cards, independent probability. I have two decks of cards, I will pull at random a single card from each deck. What is the probability that, exactly one card is a queen?
My text book is saying 24/169 after being simplified, but after my working out I get 12/169.  Can someone explain were I am going wrong?
 A: You can draw exactly one queen in two ways:


*

*The card from the first deck is a queen (this has a probability of $(4/52)$) and the card from the second is not a queen probability $(48/52)$. "And" translates to multiplication of probabilities (the events are independent), so the probability of this scenario is $$\frac{4}{52}\cdot\frac{48}{52}$$ OR

*The card from the first deck is not a queen (48/52) and the card from the second is a queen (4/52). "And" translates to multiplication of probabilities, so the probability of this scenario is $$\frac{48}{52}\frac{4}{52}$$
"OR" translates to addition of probabilities (because the events are mutually exclusive), so in total you have that 
$$\frac{4}{52}\cdot\frac{48}{52}+\frac{48}{52}\cdot\frac{4}{52}=\frac{24}{169}$$

A: First of all, you need to show me how you got that result in order that I can explain where you went wrong.
Anyway, I am showing you the correct way to approach this problem and get the correct answer.
A deck has $52$ cards and out of them, $4$ are queens.
So you can get exactly $1$ queen by choosing a queen from deck $1$ and a non-queen from deck $2$. Similarly you can get exactly $1$ queen by choosing a queen from deck $2$ and a non-queen from deck $1$.
So the probability of getting exactly $1$ queen is 
$= \large\frac{4}{52} \cdot \large\frac{48}{52} + \large\frac{48}{52} \cdot \large\frac{4}{52} $
$= 2 \cdot \large\frac{12}{13^2}$
$= \large\frac{24}{169}$
Now, as far as my intuition goes, you did not consider the $2$ different decks as different cases.
