SOLUTION: If a person draws four cards from standard deck without replacing them, what is the probability that at least one of the cards is face card ?

Question 1188314: If a person draws four cards from standard deck without replacing them, what is
the probability that at least one of the cards is face card ?
Answer by Edwin McCravy(20055) (Show Source): You can put this solution on YOUR website!

There are 3 kinds of face cards, jacks, queens, and kings, and four suits of
each which makes 3*4=12 face cards in a 52-card deck.  So there are 52-12=40
non-face cards.

When you see the words "at least one", always first find the probability of the
so-called complement event, the event of failing.  That is, we find the
probability of completely failing to get any face cards at all and then subtract
from 1 to get the probability of succeeding (getting at least one face card).

The number of ways he can fail, that is, to draw four cards from just the 40
non-face-cards, is C(40,4) = 91390 ways.

The number of ways he can draw any four cards from the entire 52-card deck is
C(52,4) = 270725 ways.

So the probability of failing to get a face card is to get one of the 91390 ways
to fail out of the 270725 ways to get any four cards.

That's a probability of failing of 91390/270725 which reduces to 1406/4165,
which is approximately the decimal 0.33757503.

So the probability of succeeding is that probability subtracted from 1 and
1 -  1406/4165 is 2759/4165 which is about 0.66242497 or 66.2%. 

Edwin

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