In Blackjack, the players and the dealer are dealt two cards each. The suits of
ID: 3354852 • Letter: I
Question
In Blackjack, the players and the dealer are dealt two cards each. The suits of the cards have no role in blackjack. Each face card is given the value of 10, aces have the value of 1 or 11, and all other cards have the value indicated by the cards themselves. Each player has the option of hitting (taking more cards) or standing (not taking any more cards). Once each player has had a turn, the dealer hits until reaching a total of 17 or higher. The players’ goal is to beat the dealer. This occurs when the player has a total higher than the dealer’s without Busting (having a total of over 21), or when the dealer busts and the player does not. Thus, if the dealer busts, then all players who have not busted win. A natural, or blackjack, is an ace-10 pair dealt before anyone hits. (Remember that all face cards have the value 10, so an ace-queen is a natural, as is an ace-10 where the 10 is actually a card numbered 10.) Naturals always win except against a dealer’s natural, in which case it is a tie.
Suppose you are playing with an infinite deck. That is, you are dealt cards with the “normal” probabilities (1/52 for any given card) which do not change as the cards are dealt. For example, if you are dealt A, then the probability that your second card is also A is still 1/52. What is the probability of being dealt a natural? How does this compare to what is the probability of being dealt a natural in an n-deck game of blackjack; that is, in a game in which n decks of cards are shuffled together?
Explanation / Answer
Probability of drawing a natural = P(Ace in first)*P(10values cards)*2 = 2*4/52*16/52
P(natural in n cards) = 2 * 4n/52n * (16n)/(52n-1)