Question

2. (10 points) Suppose you are playing blackjack against a dealer. In a freshly shuffled deck, what is the probability that either you or the dealer is dealt a blackjack? (Recall the game of blackjack is played with each player obtaining two cards at random from a regular deck. We say these two cards form a blackjack if one card is an ace and the other card is either a 10, Jack, Queen or King)

Answer 1

To find P(either you or dealer is dealt a blackjack)        
P(either you or dealer is dealt a blackjack)        
     = P(exactly one player gets a blackjack)    (We are assuming both will not get blackjack at the same time)    
         
Number of total deals to the two players         

52C2 x 50C2- 1624350

         
Number of ways exactly one player gets a blackjack         

3c 32
      
(i.e., 2C1 = choose which player to give the blackjack,         
4C1 = choose which ace to give them, 16C1 = choose which 10,J,Q,K to give them.         
Then for the other player we either give them two aces from the 3 that are left (3C2),         
give them one ace from the 3 that are left (3C1) and 1 of the 32(32C1) 2-9 cards,         
or give them two non-aces from the 47 non-aces that are left (47C2).)        
         
P(either you or dealer is dealt a blackjack)        

151040 1624350

   = 0.09299       
P(either you or dealer is dealt a blackjack) = 0.09299