What’s wrong with the following reasoning that $P(A|B)= \frac{25}{51}$? A standard deck of cards is shuffled well.
Two cards are drawn randomly, one at a time without replacement.
Let A be the event that the first card is a heart.
Let B be the event that the second card is red.
Find P(A|B) and P(B|A).
Let’s start with P(B|A):
If first card is a heart, then remaining cards consist of 25 red and 26 black(all of which are equally likely)
Therefore the conditional probability of getting a red card is $\frac{25}{25+26}=25/51$
The book(introduction to probability) then goes on to state that it’s harder to find P(A|B) in this way.
My question is why is it harder to find P(A|B) using reasoning similar to above?
Why is the following incorrect:
$P(A|B)=P(A|\text{red and heart on second})+P(A|\text{red but not heart on second draw})$
(partition event B)
$=12/51 +13/51 $ because if second draw is a heart, then there are 13-1 possible heart cards for the first draw, out of 51 equally likely cards, and then if the second draw is red but not a heart then there are 13 possible hearts for the first draw out of 51 equally-likely-to-be-chosen cards.
 A: Using intuition, knowing that the first card was red doesn't change the probability of choosing a heart that much, so it seems wrong that it could be almost double.
$P[A|B \text{ or } C]$ is not necessarily equal to $P[A|B]+P[A|C]$, even if $B$ and $C$ are disjoint.
Suppose $B$ and $C$ are disjoint, like they are in the example where $B$ means the first card is a heart and $C$ means the first card is a diamond.
$P[A|B \text{ or } C]=\frac{P[A \text{ and } \{ B \text{ or } C\}]}{P[B \text{ or } C]}
=\frac{P[A \text{ and } B]+P[A \text{ and } C]}{P[B \text{ or } C]}$
But,
$P[A|B]+P[A|C]=\frac{P[A \text{ and } B]}{P[B]}+\frac{P[A \text{ and } C]}{P[C]}$
So, these two things are not necessarily the same.
If you further assume that $P[B]=P[C]$, as in this example, then $P[B \text{ or } C]=2P[B]$ and then you will have the first is half as big as the second.
A: The problem with your reasoning is that you don't take into account the probability of the second card to be a heart or not (p = 0.5).
P(A|B) = (12/51)x0.5 + (13/51)x0.5 = 25/102.
