Blackjack Solutions

  1. The deck has a total of 52 cards. You bust if you get a card from 6 to 10 or any face card (aces won't bust, since you can count them as 1), that is, 8 types of cards times the 4 times they appear in the deck. Hence, the busting probability is 32/52 = 8/13 &asymp 0.615.
  2. The probability is 1, since whatever card you are dealt, you can count the ace as a 1 and not bust (you were dealt a soft hand).
  3. In this case we must take into account the composition of the remaining cards in the deck. The deck contains all of the cards except for an 8, a 6, a queen, and one more card we don't know (the dealer's second card). To solve our problem, we need to figure out how the dealer's second card affects the odds of drawing any particular card from the remaining deck. After giving this a little thought, it becomes clear that the probabilities are not affected; since we don't know anything about the dealer's down card, all cards not shown in the game are drawn with equal probability.

    To convince yourself of this, suppose that the dealer is initially dealt only one card, but when you hit, she first draws a card for herself and then hits you. This is clearly equivalent to the actual game situation. The probability of you busting does not depend directly on the card the dealer draws for himself, but in the card you receive. For any specific card in the remaining deck, the probability that you get it is the product of the probability that the dealer did not receive the card and the probability that you did receive the card: (48/49) · (1/48) = (1/49), just like the probability of getting any given card if the dealer were not dealt another card.

    With this, we can solve the problem. You will bust if you get a card from 8 to 10 or a face card. There are 6 · 4 = 24 of these cards in a regular deck, but the game is showing 2 of them, so there are 22 remaining. Thus, the probability of you busting is 22/49 &asymp 0.45. The fact that the dealer was dealt a card whose value you don't know does not affect your knowledge of the remaining cards in the deck and you can consider probabilities as if it was still in the deck.

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