Probability with a deck of cards:

Question

You select 10 cards randomly from a deck of 52 cards.
What is the probability that all of the cards selected are face cards (i.e. jacks, queens, or kings)?

Answer 1

The number of successes is ₁₂C₁₀ = ₁₂C₂ choose 10 face cards from the 12 in the deck.

The number of ways to choose 10 cards from the deck is ₅₂C_10.

The probability is ₁₂C₂/₅₂C₁₀

The calculation gives

(12×11/2)/((52×...×43)/10!)=4.171927873E-09 as the probability

Answer 2

You didn't sat whether each card is replaced back in the deck after it's picked.  That makes a big difference.  However, the way the question is worded ("pick 10 cards ... from a deck") makes it sound like you are picking a set of 10 cards and keeping them out -- i.e., WITHOUT replacement.  So let's assume that.

 

In each suit, there are 3 face cards, out of 13.  Since the distribution is the same in all suits, that means that the probability of picking a face card when you pick the first card is 3/13, or 12/52

 

After you've picked the first card, if it's a face card then there are only 11 faces cards left, out of a total of 51.  So the probability of the next card also being a face card is 11/51.

 

Keep going like that for all ten picks, and you end up with the following probability:

 

12    11    10            3

--- x --- x --- x ... x ---

52    51    50           43

 

Another way to express this is to see that the numerator is 12! / 2! and the denominator is 52! / 42!

 

So it would be (12! / 2!) / (52! / 42!) = (12! 42!) / (2! 52!)