Dealer Probabilities in Blackjack
Dealer probabilities in blackjack affect how you should play your hand. Sometimes, it's the best strategy to stand on a low total, especially if you think the dealer has a good chance of busting when their turn comes. Part of the house edge in blackjack comes from the fact the player must play first, which exposes the player to a chance of busting out. Even if the dealer busts, too, in that situation, the player has already lost his bet. This is mitigated by the fact casinos require the dealer to act according to specific rules, so they can't stand pat on a 16 after you've busted. For this reason, it's sometimes a good idea to get out of the way and let the dealer take the risks.
This article discusses how to use the dealer's upcard as a key to your play. The probability of the dealer busting when the house shows a certain upcard affects your decisions. So does the probability of the dealer holding a 21 (based on her upcard). Several key blackjack rules (peek rules, surrender, insurance) involve the dealer's upcard possibilities, so take this as an indication of how pivotal the experts (game designers) view this piece of information.
The Dealer's Odds of Having a Blackjack
The probability of a dealer hitting a blackjack when she’s holding an ace is nearly one in three. For instance, in a game with one deck, where the dealer stands on a soft 17 (not a common occurrence), the odds of hitting blackjack are 31.37%. This chance goes down a bit in a game with 2 decks (31.07%), and it goes down even more in a game with 4 decks (30.91%). The impact of the ace being out of the deck is watered down by the inclusion of more decks. In this theoretical scenario, 16 out of 51 cards would produce a blackjack in a single-deck scenario, while 16 of out 51.5 cards would produce the same result in a two-deck scenario.
Some players make the mistake of assuming an upcard with an ace or a 10 is the same, since each offers the chance at a blackjack. When a 10 is showing, the dealer has much fewer outs (4 in 51), so your chances are much better (7.84%).
Soft 17 Probabilities
Let's switch to a dealer hitting on a soft 17, instead, since this is a much more common game rule. The probabilities are counter-intuitive in this game. When the dealer is showing an ace, the chances of making 21 are 9.26%, while the chances of making a 21 with a 2-card showing is 12.19%. The reason for this is the cards required which stop the hand when the player holding an ace reaches 18, 19, or 20. The chances of a player busting when they hold an ace are much lower than they are for the 2 through 6 card. For the ace, the probability of a bust is 20.45% for an ace, 35.63% for a two, 37.89% for a three, 40.60% for a four, 42.99% for a five, and 43.78% for a six. At this point, the chances of busting go down significantly, with every card between a 7 and 10 less than 25%.
This is one of the reasons strategy charts call for caution when you have a 12-17 and the dealer holds the lower cards, because basic strategy calls for you to get out of the way of yourself and let the dealer bust. It reminds me of a historian who talked of a defeated nation which prospered in the afterwar years, because its leaders "did nothing and did it well". Sometimes the best strategy is to let your opponent make a mistake. Many gamblers are in the game for the action, so pursuing a policy of inaction goes against their instincts.
Basic Strategy and the Upcard
Profiting from the dealer probabilities in blackjack is just one reason to know basic strategy before you ever sit down at a blackjack table. If you study the theories behind the numbers, you might reinforce the underlying logic of basic strategy, which makes it easier to go against your intuition. Look at an analysis of an infinite number of hands using basic strategy, cross-referenced against the dealer's upcard. These charts exist on the Internet and should give you an idea of the numbers the dealer faces. Though it's always a good rule of thumb to keep in mind the 31% probability (or somewhere around that) of a blackjack, it's also important to keep a broad idea of the chances that the casino's hand will bust, too.