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How do you determine the different odds in Blackjack? For example, what would be the difference in odds from using 1, 2, or 3 decks? Also, what would be the difference in odds if you shuffle the deck before each hand instead of after the cards are used up?

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Generally, as you use more decks you are approaching drawing with replacement instead of without. The chance of getting a blackjack with one deck is 2(4/52)(16/51), with the 2 being whether you draw the ace first, 4/52 drawing an ace, and 16/51 drawing a 10 count. If you have a huge number of decks this becomes 2(4/52)(16/52). If you don't shuffle before the hand, it is impacted by the cards you have seen go by. If all the aces have already been drawn, the probability is zero.

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Edward Oakley Thorp went into depth on this subject with his book ${\bf Beat \,\, the \,\, Dealer,}$ which demonstrated mathematically how the house advantage can be overcome in blackjack by card counting.

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