This might be trivial, but I just want to make sure I don't struggle on the basics.
I am confused about this:
"Draw one card from a standard deck of playing cards. Let’s examine the independence of 3 events ‘the card is an ace’, ‘the card is a heart’ and ‘the card is red’. Define the events as A = ‘ace’, H = ‘hearts’, R = ‘red’." (Source)
With $P(A) = \frac{1}{13}$, $P(H) = \frac{1}4$ and $P(R) = \frac{1}2$
I get $\frac{1}4$ for $P(H|R)$. So this basically says that $P(H)$ and $P(R)$ are independent, since H and P are independent iff $P(H|R) = P(H)$.
I know that that's not correct from the source, so now I am wondering how I can then calculate the conditional probability for dependent events and how I would know beforehand. Why is $P(H|R)$ actually $\frac{1}2$?
I feel like I am missing one bit of the puzzle or overcomplicating things, so any help is appreciated!
Best regards, Sam!
$P(H|R)$ means you already know the card is red, so your total pool is $26$ red cards. from these $26$ red cards, you want to know the probability for selecting a heart, so there are $13$ heart cards, therefore, you have
$$P(H|R)=\frac{13}{26}=\frac{1}2$$