A poker hand consisting of 4 cards is dealt from a standard deck of 52 cards.
Find the probability that the hand contains exactly 2 face cards.
There are 4 Kings, 4 Queens, and 4 Jacks in a standard 52-card deck. That's 12
face cards. The other 52-12 = 40 cards are non-face cards.
First we find the total number of possible 'successful' card-hands, i.e., those
that contain exactly 2 face cards (and therefore exactly 2 non-face cards).
The number of pairs of face cards are:
12 face cards CHOOSE 2 = 12C2 = (12∙11)/(2∙1) = 66 ways
The number of pairs of non-face cards are:
That's 40 non-face cards CHOOSE 2 = 40C2 = (40∙39)/(2∙1) = 780 ways
For each of the 66 ways to choose two face cards, there are 780 wats to choose
two non-face cards to put with them to make a 'successful' 4-card hand
That's 66∙780 = 51480 successful 4-card hands.
Next we find the total number of 4-card hands, 'successful' ones and 'failures'
together.
That's 52 cards CHOOSE 4 = 52C4 = (52∙51∙50∙49)/(4*3*2*1) = 270725 possible
4-card hands.
So the probability is 51480/270725 = 0.1901560624
Answer: Very nearly 19% of the time you'll be dealt a hand that contains
exactly 2 face cards and exactly 2 non-face cards.
Edwin
