Probability Another Playing Card question -- twist I completely understand the questions of probability of drawing 2 cards without replacement -- getting a heart or a face card.  I add the probability of getting a heart with probability of getting a face card and subtract the doublecounting of the face cards that are hearts.
But what if the question reads, "what is the probability of getting a heart and a face card?  This would be a multiplication problem of the 2 probabilities.  I understand that the probability of getting a heart is 13/52, but now how do I account for the conditional probability of the probability of the face card given I have drawn a heart on the first card?
 A: We can interpret the problem in two ways.  In what follows, we denote a card that is neither heart nor face card by $N$, a card that is a heart but not a face card by $H$, a card that is a face card but not a heart by $F$, and a card that is both by $B$.  By the usual definition of "face card," there are $3 B$, $9 F$, $10 H$, and $30 N$.  If your definition varies, you may adjust the analysis accordingly.
We preface by pointing out that there are $(52)(51) = 2652$ different ways to choose the first two cards (differentiating the first and second cards).
(a) The strict interpretation: The heart and the face card must be distinct.  The combinations that satisfy this interpretation are $BF, FB, BH, HB, FH, HF$, and $BB$.  The count of these combinations is
$$
(3)(9)+(9)(3)+(3)(10)+(10)(3)+(9)(10)+(10)(9)+(3)(2) = 300
$$
So the probability under this interpretation is $300/2652 = 75/663 \doteq 0.11312$.
(b) The loose interpretation: The heart and the face card may be the same card.  The combinations that satisfy this interpretation are all of the above, and in addition $BN$ and $NB$.  The count of these combinations is
$$
300+(3)(30)+(30)(3) = 480
$$
So the probability under this interpretation is $480/2652 = 120/663 \doteq 0.18100$.
