This discussion will look at:
The rules of Black Jack are quite simple, although the computations that determine the best strategy is complex. The object of the game is to total closer than the dealer to 21 without going over. You are given two cards and you get to see one of the dealer's two cards. You can receive an additional cards until either you decide that you are close enough to 21 or your total is greater than 21 in which case you lose your wager. An A counts as 1 or 11 and the J, Q, and K count as 10. There are three more rules that allow you to better your odds. After receiving your first two cards, you can "Double Down" which means that you double your bet and receive exactly one card. This is a good idea if you have a hand that requires exactly on hit and when you are relatively confident that your hand is better than the dealers. A second special rule is the "Split." If your first two cards are a pair, you can put an amount of money equal to your wager and play two hands each with one of the two cards. The third special rule is the "Insurance" bet. If the dealer has an ace showing, then you can put down additional money equal to half your wager. If the dealer has Black Jack (a 10, J, Q, or K as the face down card) then your Insurance wager wins 2 to 1. The net result is that you do not lose any total money. If the dealer does not have the Black Jack, then you immediately lose the insurance bet.
Below is a table of strategy for Black Jack. "H" means hit, "D" means double down, "S" means stand, and "SP means split. Many casinos will not allow the player to double down on any two cards. If you are not allowed to double down when there is a "D", you should hit instead.
For example if you are dealt an Ace and a 4 and the dealer has a 6 showing you should double down.
Some casinos offer a few other special side bets. One such bet is the Royal Match. The Royal Match involves wagering that your first two cards will be the same suit. If they are, then you get a payback of 3 to 1. If the two cards are a K and Q of the same suit, then you get a payback of 10 to 1. Let's compute the expected value of this bet assuming that you are playing at a single deck table.
To compute the probability of getting a K and Q of the same suit, notice that for the first card, 8 of the 52 cards will give you a chance. Once the first card is chosen, only 1 of the 51 cards that remain will give give you the Royal Match. Hence the probability is
P(Royal Match) = (8/52)(1/51) = 0.0030166
To compute the probability that your first two cards are the same suit, note that this means that the second card is the suit of the first card. After the first card is dealt, there are 51 cards left, 12 of the same suit as the first card. We then need to subtract the Royal Match probability so that we do not double count it. Hence this probability is
P(Same Suit) = 12/51 - 0.0030166 = 0.232277
Now to compute the probability that you lose, just subtract the above probabilities from 1.
P(Lose) = 1 - .0030166 - 0.232277 = 0.764706
Since the payoffs are 3 to 1 and 10 to 1, we can compute the expected value if $1 is wagered.
EV = (10)(0.0030166) + (3)(0.232277) + (-1)(0.764706) = -.0377
This means that if you make this wager many times, you can expect to lose on average about 4 cents per game.
We will show that calculation of the insurance bet in a few different situations. In each of these assume that you are playing at a single deck table.
You are dealt a 10 and Q. Should you buy insurance?
We will compute the expected value. To win the insurance bet, the dealer must have a 10, J, Q, or K as the down card. Since you have a 10 and Q there are only 14 cards left that make your insurance bet win. There are 49 unseen cards left in the deck (52 minus your 2 and the dealer's Ace). Hence the probability of winning the insurance bet is
P(Win Insurance) = 14/49 = 2/7
The probability of losing is
P(Lose) = 1 - 2/7 = 5/7
The expected value for a $1 wager is
EV = (2)(2/7) - (-1)(5/7) = -0.143
In this situation you can expect to lose an average of about 14 cents per try.
Situation 2 (Card Counting)
Card counting is the process of keeping track of the cards that you have already seen. Here is a simplified example. Suppose you and your buddy are gambling together. You hold a 7 and a 9 and your buddy holds a 2 and a 5. Again assume that the dealer has an ace showing. Now there 47 are unseen cards and 16 of them will make your insurance bet win. The probability of winning the insurance bet is
P(Win Insurance) = 16/47
and the probability of losing is
P(Lose) = 1 - 16/47 = 31/47
The expected value for a $1 wager is
EV = (2)(16/47) + (-1)(31/47) = 0.021
In this situation you can expect to win an average of about 2 cents per try. This is the power of counting cards. It allows you to adjust your betting and playing strategy base on whether your expected value is positive or not. In the old days (about 15 years ago) expert card counters could win thousands of dollars in Black Jack; however, casinos have become savvy and now adjust the rules for single deck tables. For example, Harrahs Tahoe will not allow you to double down unless your first two cards sum to 10 or 11. This small change in the rules changes the total expected value from positive to negative.
For the beginning who has decided that it is worth losing $100 or so on average at a sitting at a Black Jack table, I would recommend taking the table of strategy to the table with you and playing at a 6 deck table. The key is to treat gambling as entertainment rather than a way to make money. Think of gambling as an expensive movie, where you are paying money to be entertained.
The Insurance bet is a minor reason to count cards. The larger reason has to do with the player's advantages. The dealer's advantage is if both the dealer and the player bust, then the house wins. The player has several lesser advantages. The first is that the player receives 21 on the first two cards then the payback is 1.5 to 1, whereas the dealer does not receive the extra cash. The other player's advantage is that the player can double down or split and can make a decision of hitting or standing. The dealer cannot double down or split and has predetermined hitting and standing rules. If you look at the table of strategies, most are based on the player and or the dealer receiving a 10 card. Hence if there are many 10 cards left and not so many lower cards left, then the player's advantage is accentuated. This is the basis of the counting system. The "count" for each card is listed below.
2 through 7 Count of +1
8 and 9 Count of 0
10,J,Q,K Count of -1
A Count of -2
If the total count is positive then the player should wager more money and if the total count is negative, the player should wager less money.
After you become familiar with the basic counting system you can try two modifications. The first is that you should always keep track of how many aces have gone by. This helps with the insurance bet. For example, if the count is 0 but all four aces have been seen, then insurance is a good bet.
A second modification has to do with the smaller cards. It turns out that it is better to have twos and sevens left than to have fives and sixes left. If you are playing at a full table, then there are two rounds per deck. If after the first round you have seen more fours and fives than twos and sevens, add 1 to the count. If you have seen more fours and fives than twos and sevens, subtract 1 from the count.
The cards are dealt quickly, so card counting takes full concentration. Do not mix card counting with alcohol. If you want to drink, play at a table with 6 decks. The casino will let you take the piece of paper with the basic strategy to the table with you, but will not let you use a calculator or other electronic device.
Always decide how much you are willing to spend before you begin. Never go beyond this number. If you find yourself pulling out more cash to get back what you lost, then you have a gambling problem and should call the National Council on Problem Gambling at 1-800-522-4700.