Thursday, November 23, 2006

Statistical Probabilities in Blackjack

All casino games favor the house and the “Law of Large Numbers” guarantees that you will loose in the long-haul. However, there are some games that approximate a fair deal:

Single deck black jack – If you learn to accurately count cards (a perfectly legal practice as long as you don’t use artificial appliances), and recalculate probabilities you can win big money. It’s not hard to develop a mental mnemonic for card counting (Hint: remember than ten-value cards comprise 30% of the deck).

Blackjack probabilities

Blackjack is a non-zero sum, two person game (you play against the dealer), with well-defined decision rules. If you are not able to count into a single-deck, the rules of blackjack are very straightforward, and fully quantifiable. I’ve won several blackjack tournaments, and it amazes me how many people do not fully understand the decision rules for playing blackjack.

Many people only know the basic rule for blackjack (hit 16, stand 17), but many people forget that the dealers up-card also influences your chances of winning. The most common value of cards in the deck is a ten-value (16 cards, or 30% of the deck), so many experienced blackjack players will assume a ten for the dealers hidden card.

The probability of the dealer busting on the first draw

More sophisticated players will factor the probabilities of the dealer’s down-card, assuming at there is a 30% change that the down-card is a ten-value. Using this maximum assumption of a ten down-card, you can develop a set of hit-or-stand rules.

For example, if the dealer has a six up-card, then the dealer probability of busting is over 18%, and the black jack rules state that the dealer must hit up to 16.

Note: These probabilities have been simplified for easy understandability. Sophisticated players will mentally adjust these probabilities based on the known cards in their hand.

Dealer with a deuce up = Dealer busts with a face-card or a ten =
(16/52)*30% = 9%

Dealer with a three up = Dealer busts with face-card, ten or nine =
(20/52)*30% = 11%

Dealer with a four up = Dealer busts with card greater than seven =
(24/52)*30% = 14%

Dealer with a five up = Dealer busts with any card greater than a six =
(28/52)*30% = 16%

Dealer with six up = Dealer busts with any card greater than five =
(32/52)*30%) = 18%

When it’s safe to hit

Remember, it’s always safe to take a hit if you have eleven or less (always go down for a double with eleven), or any “soft” combination (where one card is an ace), up to seventeen.

When to stay?

As we see above, if the dealer has an up-card of a six, the dealer has an 18% chance of busting on their draw, and you must compare their probability of busting with your probability of busting.

For example, if you and the dealer have a total of 16, both you and the dealer have an 18% chance of busting, and many players (me included) will stand and let the dealer draw out.

Of course, it’s the random nature of blackjack that makes it exciting (plus the challenge of single-deck blackjack, where you get a change to practice remembering the played cards), so these probabilities only serve as guidelines. Remember, anything can happen with the dealer has a five-deck shoe (nobody but Rain Man can count into a five-deck shoe), but these basic probabilities still apply.