17 July 2009

Blackjack Probability

My life experience, let me show you them.

At one point in my moderately shady past I worked as dealer on a Louisiana riverboat casino. I’ll take a moment here to let you bask in the fact that my past rocks. Done? Okay, this is to anyone who has ever played Blackjack – I would like to offer you a crash course in luck v. probability vis-à-vis card counting.

The difference between blackjack and the games I will go into later is simple, in blackjack your odds are constantly changing. Discarding suits, and I know of at least three variations on Blackjack that offer varying odds on certain suited sets of cards, there are thirteen different cards you can pull [A2345678910JQK]. Assuming a full deck, you have approximately a 7.6% chance of pulling any given card. However, in the game of blackjack, there are four cards that are considered to be ‘ten cards’ [10JQK]. All of these have the same value on the table and are interchangeable within the gameplay. You could take out all of the Jacks, Queens and Kings in a deck and replace them with 10s and the game would play the same. Thus, while the probability of drawing any single card is 7.6%, the probability of drawing a ten card is in fact 30%.

Let’s play with that for a moment. Let’s say I have a table with six players who all, on the first hand, receive natural twenties (two ‘ten cards’). I also receive a natural twenty. On the table there are now 14 ten cards. A six deck shoe (common for the area I dealt in) contains 312 cards. One is ‘burnt’, or discarded if you will, before the first hand is dealt. At the end of the first hand there are 297 cards left in the shoe. The shoe stared with 96 ‘ten cards’ and now contains 82. After the first hand, the total probability of pulling another ten card is now 27%. Until more cards are pulled, the probability of pulling a ten has dropped three percent. After one hand. If that doesn’t make you pause for a moment then I don’t know what will.

So how do we get back to a 30% probability of pulling a ten card? It seems easy. Pull 14 non-ten cards to balance it out. If we do this, however, our probability only rises to just shy of 29%. In fact, to return to a 30% probability, you’d need to pull nearly thirty non-ten cards from the shoe.

The above math is what card counters are attempting to take advantage of. As cards are pulled, the odds for that type of card appearing again will fluctuate in relation to other cards pulled. By keeping a count of what cards have already been used, you can make a guess at what cards are likely to appear.

So how do casinos combat this? The easiest way is in the cut. Ever notice the dealer doesn’t deal all the way to the last card? Instead, they deal to a special yellow cut card, finish the hand, then shuffle all decks. At my casino, procedure was for the dealer to cut at least a deck and a half to two decks from the end of the shoe. If you had a suspicion the player may be counting cards, you cut straight in the middle, three decks from either end. While this doesn’t change the overall odds, at any given time a significant portion of remaining cards will never be used or seen. In theory, with a deck where 156 of the cards will never be used (a halfway cut), all ten cards and aces could be on the other side of the cut card. While the odds of pulling one of these will grow closer and closer to 100%, knowledge of this is ultimately useless as there is literally no way to access these cards.

The more cards a counter sees, the better informed they are as to what their next card will be. If a dealer dealt to the last card, a person card counting would know with 100% probability what that last card will be. (This is actually giving counters a little too much credit. For the most part, the systems used merely give the counter a ‘best guess’ at what the nature of the next card will be [High/ Low/ Neutral] rather than the specific type of card [A23456789‘10’].)

So the moral of the story is, if you ever see a dealer cut a shoe halfway then they or their pit manager suspects a card counter at the table. And honestly, trying to count into a six deck shoe while nearly a third of the deck remains unknown is a little silly. You only gain a measurable advantage for the last two or three hands. Stick to basic strategy, you’ll probably work out better in the long run.


Probability of any type of card: 1/13 or .076923… (repeating)

Probability of 'ten card': 4/13 or .307692… (repeating)

Probability of 'ten card' after first hand: 82/297 or .276094… (repeating)

Probability of 'ten card' after fourteen non-'ten cards are removed after the first hand: 82/283 or 0.289752 (non-repeating)


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