help????????

Question

You draw 3 cards from a standard deck of 52 playing cards. What is the probability that you get at least 2 face cards?

Answer 1

There are 52 cards in a deck of cards, and 12 face cards (4 suits, King, Queen, and Jack in each suit).

 

Scenario 1:

the first card is NOT a face card

 

the second card must be a face card in order for the scenario to continue, and the odds will be 12 out of 51 = 0.2353

 

the third card will have odds of 11 face cards out of 50 cards = 0.2200

 

the combined probability will be 0.2353 x 0.2200 = 0.0518

 

Scenario 2:

the first card is a face card

on the first draw, the odds will be 12 out of 52 = 0.2307

 

on the second draw, the odds will be 11 out of 51 = 0.2156

 

the combined probability will be 0.2307 x 0.2156 = 0.0497

 

 

Answer 2

You can solve this problem by making a probability tree, which will tell you the probability of drawing any particular sequence of cards from the deck.  

 

Let F denote a face card and let N denote a non-face card.  Note that a standard deck contains 12 face cards (J-Q-K in each of 4 suits) and 40 non-face cards.  So on the first draw you have  

 

P(F) = 12/52

P(N) = 40/52  

 

Note that these 2 probabilities sum to 1.  This is a very important property of probability trees, because we have listed all the possible outcomes (this means that we have covered the entire sample space) and there is no overlap between the possible outcomes (this means that the outcomes are disjoint).  

 

Now consider the second draw.  There are 2 possibilities - the first draw was either F or N.  If the first draw was F then there are now 11 face cards left in the deck out of 51 remaining cards.  If the first draw was N then there are still 12 face cards left out of the 51 remaining cards.  So you can determine the following probabilities after drawing 2 cards  

 

P(FF) = (12/52)*(11/51) = 132/2652

P(FN) = (12/52)*(40/51) = 480/2652

P(NF) = (40/52)*(12/51) = 480/2652

P(NN) = (40/52)*(39/51) = 1560/2652 

 

Note that again these 4 probabilities sum to 1.  Also note that 1 of these sequences has exactly 2 face cards, 2 of these sequences have exactly 1 face card, and 1 of these sequences has exactly 0 face cards.  I will leave it to you to continue to the third branch of the tree by considering the third draw.  You will need to consider all 4 of the above possibilities and take into account how many face cards and non-face cards are left in the deck out of the 50 remaining cards for each possibility.  You will end up with 8 possible sequences and a corresponding probability for each sequence.  The sum of all the probabilities should be 1.  You can count how many of the sequences have at least 2 face cards and, since the outcomes are disjoint, you can simply add up the individual probabilities to get the total probability of getting at least 2 face cards.