Probability of deck of cards

Question

You created a new deck of cards, using a standard deck of 52 plus a second set of spades (ace,2-10,jack,queen,king). From this new deck, 7 cards are picked at random. What is the probability that none of the 7 cards is a face card? Face cards are: jack, queen, king, and ace. Show all work.

Answer 1

Hi Chris.

 

So with this newly created deck we have 65 cards.

We have 15 face cards (3 x 5)

 

So the probability of picking a face card from the full deck is 15/65

And the probability of not picking a face card from the full deck is 50/65

But we pick 7 cards.  We need to calculate this without replacement.

 

Probability that the first card is not a face card = 50/65

Probability that the 2nd card is not a face card = 49/64

Probability that the 3rd card is not a face card = 48/63

Probability that the 4th card is not a face card = 47/62

Probability that the 5th card is not a face card = 46/61

Probability that the 6th card is not a face card = 45/60

Probability that the 7th card is not a face card = 44/59

 

Now

 

 50     49     48      47     46     45     44         503,417,376,000

---- • ----- • ----- • ----- • ----- • ---- • ------- = -------------------------------     = .14347

 65    64      63      62     61     60    59         3,508,800,422,400