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Dependent Events in the game of Blackjack
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Less Blackjack not your bag? That's easily calculated, but it varies based on how many decks are being used. For this example, we'll use one deck.
To get a blackjack, you need either an ace-ten combination, or a ten-ace combination. Order doesn't matter, because either will have the same chance of happening.
You have four aces in the deck, and you have 52 total cards. There are 16 cards in the deck with a value of ten; four each of a jack, queen, king, and ten. You want to know if one or the other is going to happen, so you add the two probabilities together. You're playing in a single deck blackjack game, and you've seen 4 hands against the dealer.
In all 4 of those hands, no ace or 10 has appeared. You've seen a total of 24 cards. There are only 28 cards left in the deck. Again, you could get a blackjack by getting an ace and a ten or by getting a ten and then an ace, so you add the two probabilities together. This last example demonstrates why counting cards works. The deck has a memory of sorts. If you track the ratio of aces and tens to the low cards in the deck, you can tell when you're more likely to be dealt a blackjack.
Since that hand pays out at 3 to 2 instead of even money, you'll raise your bet in these situations. The house edge is a related concept.
It's a calculation of your expected value in relation to the amount of your bet. Expected value is just the average amount of money you'll win or lose on a bet over a huge number of trials.
Using a simple example from earlier, let's suppose you are a 12 year old entrepreneur, and you open a small casino on the street corner. You allow your customers to roll a six sided die and guess which result they'll get.
Over every six trials, the probability is that you'll win five bets and lose one bet. That's how the casino does the math on all its casino games, and the casino makes sure that the house edge is always in their favor.
With blackjack, calculating this house edge is harder. After all, you have to keep up with the expected value for every situation and then add those together. Luckily, this is easy enough to do with a computer. We'd hate to have to work it out with a pencil and paper, though. It depends on the game and the rules variations in place.
It also depends on the quality of your decisions. If you play perfectly in every situation—making the move with the highest possible expected value—then the house edge is usually between 0. Even for the gambler who ignores basic strategy, blackjack is one of the best games in the casino.
You can use this information to estimate how much money you're liable to lose or win per hour in the casino. Of course, this expected hourly win or loss rate is an average over a long period of time. Over any small number of sessions, your results will vary wildly from the expectation. The conditions under which you play blackjack affect the house edge. For example, the more decks in play, the higher the house edge.
If the dealer hits a soft 17 instead of standing, the house edge goes up. Getting paid 6 to 5 instead of 3 to 2 for a blackjack also increases the house edge. Luckily, we know the effect each of these changes has on the house edge. Using this information, we can make educated decisions about which games to play and which games to avoid. These are just some examples. There are multiple rules variations you can find, some of which are so dramatic that the game gets a different name entirely.
Examples include Spanish 21 and Double Exposure. The composition of the deck affects the house edge, too. We touched on this earlier when discussing how card counting works. But we can go into more detail here. Every card that is removed from the deck moves the house edge up or down on the subsequent hands. This might not make sense initially, but think about it.
If you removed all the aces from the deck, it would be impossible to get a 3 to 2 payout on a blackjack. That would increase the house edge significantly, wouldn't it? These percentages are based on a single deck. If you're playing in a game with multiple decks, the effect of the removal of each card is diluted by the number of decks in play. Looking at these numbers is telling, especially when you compare these percentages with the values given to the cards when counting.
The low cards have the most dramatic effect on the house edge. That's why almost all counting systems assign a value to each of them. The middle cards have a much smaller effect. Then the high cards, aces and tens, also have a large effect. The most important cards are the aces and the fives.
Each of those cards is worth over 0. That's why the simplest card counting system, the ace-five count, only tracks those two ranks. You can also look at the probability that a dealer will bust based on her up card. This provides some insight into how basic strategy decisions work. Perceptive readers will notice a big jump in the probability of a dealer busting between the numbers six and seven. They'll also notice a similar division on most basic strategy charts.