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The Mathematics Behind Blackjack

Most players of the game of blackjack learn that it is possible to gain an advantage over the casino by keeping a mental count of the cards dealt. This is called card counting. Excellent players can achieve an edge over the house, which amounts to about 1% on average, depending on the game’s rules and how skillfully they play. The advantage is only achieved under certain circumstances, but it can give players a competitive edge over even one dealer with whom they are playing.

This article will discuss the mathematics behind blackjack, such as the theoretical return to player (RTP) and how you can count cards to beat the system.

In blackjack, the player is trying to get as close as possible to 21 without going over. They stand at 20, and any cards totalling over this score will cause them to lose their bet. On the other hand, the house has a target of 17 or higher for making its profit margin. Blackjack games almost always have a house advantage of 1%. How exactly does this work?

The answer lies in the way that card decks are created. The deck has 52 cards, including Aces and face cards but not Jokers (their value is 0). If one were to shuffle a deck of cards thoroughly, they would be giving each card an equal number of chances of being the next card to be drawn. If one draws a new card every second, then there are 52/60 = 0.8333333333 or approximately 84% chance that each card will come up in, on average, 20 hands (1 hand is equal to 1 deal). This means that 16% of the time, each card will be drawn in approximately five hands. So while every card has an equal chance of being dealt out, some items will appear more often than others because when humans modify a distribution to become less random and more unstable, there are bound to be discrepancies. Thus the theoretical RTP in blackjack is 54%.

Card Counting: The Mathematics Behind It

Card counting is the most helpful technique for gaining an advantage at blackjack and one of the oldest. To keep a count, you must first determine which cards have already been dealt from the deck while other players are playing their hands. Once you know what has already come up in the game, you can use a moving average to predict the likelihood that the following cards will be dealt.

How Card Counting Works

When card counting works best, the deck has become rich in ten-value cards (10, J, Q, K). At that time, the player can bet more money and increase their expectation of winning (or the size of their advantage).

Here’s an excellent, straightforward explanation of card counting: A deck with more tens than usual will have lower cards in it. This means that there are fewer lower cards left in the deck than higher ones. But since the cards are indistinguishable, you can’t tell. So players using this strategy will bet more money on hands with lots of high cards – and less on those lower in value.

When a card is removed from a deck, it is impossible to say how many tens the deck holds. But there are ways to estimate this figure without holding up the game.
There are 52 cards in a deck, and there are two jokers, which makes 104 cards. Since the player typically has to stand on hands with a value of 17 or more, this leaves 48 cards from which to draw (the remaining six are disregarded because they’re low value). So we can say that there are 48/104 = 0.4734826235 or approximately 47% ten-value cards in the deck. So there are 47% more tens than usual and 53% fewer low-value cards. That means high-valued hands have a greater chance of winning or being dealt out.

Moving Averages in Blackjack

There are many different forms of moving averages, but we will use a simple one that can be calculated by adding up all the numbers from 0-10, then taking away 1/2 of them. If you do this and then add on the removed number, you obtain a sequence that starts with values of 2/3, 3/5, 5/8. The average of this is 2.75.

In the same way, every other card in the pack has an average value of 1/(52-1) = .2658139535 or 26.58%. (These values are derived from the fact that there are 26 suits, 13 numbers and 2 Jokers in a deck of cards.)

The Mathematics Behind Blackjack - Conclusion

If you sum up the values of all the cards that are left in a deck, it equals 52/104 = 0.48333333333 or approximately 48%.

So if there were 47% ten-value cards remaining and 26/52 of them (0.51875) are tens, then there is a 47% chance that the next card will be higher than a ten. And there is also a 48% chance that the next card drawn will be much lower in value than the remaining 26/52 of tens, which takes us to about 23%.

An ace only counts as one, so if you draw an ace, have a 47% + 23% = 70% chance of drawing another ten-value card.

The good news for us is that since we know a lot of cards are likely to be tens and not low value, we can bet more money on our own hands and increase our chances of winning a game.