Blackjack – Do you think you can win at blackjack? What is the best way to play? In this article, we solve these questions and more with the help of mathematics.
The goal is to get 21 with our cards or a higher value than the dealer without going over the numbers.
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After the bets are placed, each player is dealt 2 face up cards and only one is the dealer.
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Players must say based on their cards and the dealer whether they want to hit, stand, fold or split the hand.
All cards have a numerical value, that is, from 2 to 10, except for the numbers, which are worth 10, and the ace, which for our purposes can be worth 1 or 11.
The payouts are made simultaneously and we can get 21 with more than two cards, but it will only count as blackjack if we have two.
House can only hit or raise, and it only hits when his hand is below 17, so if he reaches or exceeds that value, it automatically saves.
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If the dealer goes over 21, players still at the table win their bets, and in the event of a tie, the player gets their bet back.
American blackjack has several different rules that allow you to change whether the dealer hits or how players can split their hands. Also, a second face up card is dealt and you can check it to see if its first card gives you a blackjack chance. If a 21 is rolled, players lose their bet before playing. This speeds up the game and also reduces the home advantage.
First, we can hit until our total is above 21, depending on the card we are dealt, we will be closer to 21, otherwise we will lose the bet if we exceed it.
To double our bet, we need a hand that must be 9, 10 or 11 and can only do this at the beginning of the turn. If you double your bet, you can only get one more card.
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To double we need a hand that must be 9, 10 or 11 and you can only do this at the beginning of the round. If you double your bet, you can only get one more card
If our cards are of the same value, we can split each card into different hands to play them at the same time, but this also requires adding an initial bet. You should also know that as part of a split hand, even if we add 21 with the next card, it is not considered blackjack.
After splitting, most casinos allow you to double your new hands, which gives us a certain advantage. (If it is not allowed, the house advantage increases by 0.12%).
If the card revealed by the dealer is an ace, we can bet for insurance if we believe the house will blackjack with the next card. If we have gambled and finally succeeded, we will be rewarded.
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And finally, walk away, which is only possible in American blackjack, and if the player walks away, he only loses half of his bet.
–3 to 2 (three chips, for each two-chip bet, or one-and-a-half for each bet) if we beat the dealer at blackjack (provided there is no tie). In Las Vegas American Blackjack, you pay 6 to 5.
In the case of American blackjack, it is also common that if the player has a blackjack and the dealer has an exposed ace, they are offered a 2 to 1 chance to cash the blackjack (instead of the usual 3 to 2) as an alternative to the insurance bet.
After that there are only 4 cards that are aces so if we want to get blackjack we need one but…what are the odds?
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Blackjack probability is the number of ways we must get 21 with 2 cards from all possible combinations to draw 2 cards from the deck.
In other words, favorable cases are among possible cases, which are not the same as related probabilities.
To get 21 from 2 cards, we need an ace and a 10. And the deck has 4 aces and 16 tens (between the face cards and the 10 value cards).
On the other hand, all combinations of 2 cards, combinations of 52 elements taken 2 at a time (where the order does not matter and cannot be repeated),
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4.16 / C52, 2 which is equal to 64/1326 or 4.8% chance of getting blackjack (which makes sense since we know we always need an ace to play blackjack and only 4 in the deck).
In the case of a multi-deck game, each shape or number holds the same ratio, since it is the same 4 aces in 52 cards as 8 in 104. But the probability of a certain result is slightly different, since drawing a card from a deck of 104 has less impact than drawing a card from a deck of 52. Therefore, each game is studied with different probabilities.
To calculate the probability of going over 21, we need to calculate the probability of going over any hand and for that we need to analyze the hand.
As we can see, initially 4 times out of 13 we go above 21. This is about a 30% chance, which means that at 12 we only go above 21 30% of the time.
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(The calculation of this probability is simplistic because we really have to take into account that the cards that add up to 12 are slightly different from the probability of the card we asked for, but the difference with this calculation is not very large, and it is also much more difficult to study and understand).
Also note that if our total is 11 or less, we can never go over 21 with another card, so the probability is 0.
As we just saw, if we don’t want to lose most of the time, we have to hit every time our hand is less than 14. The question is… If we limit ourselves to 14, do we have a chance to win?
Given that the dealer always takes a card up to 17 or higher, staying under 17 will only win you if the dealer wins.
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In other words, unless the dealer rolls a 21, his hand will always be worth 17 or more, so players who had a hand less than 17 will lose their bet.
Well, according to a mathematical study done, the dealer’s odds of exceeding 21 depending on the card he has (and his strategy) are as follows:
We can see that for the high card the probability of winning is quite low, and the ace is the card that gives the biggest advantage to the house.
For a low card, the probability is higher, but note that in no case does it exceed 50%
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If we average them all, we come to the conclusion that the dealer hits over 21 28.35% of the time (368, 67/13), which means that the remaining 71.65% of the time, if you are under 17, you lose your bet.
As we have just seen, it makes no sense to set the limit lower than the dealer’s limit, as in most cases we will lose.
If you notice that the dealer always waits for the player to finish his turn before continuing the game with him. This means that if the player goes over 21, they lose their bet no matter what the dealer does. So if the dealer also goes over 21, it’s the player who loses anyway because he was eliminated earlier. This makes the house win in one case, which is effectively a draw and this happens 7.9% of the time you play. That’s about an 8% edge over the player, which eventually drops to 5.6% when you consider that blackjack pays 3 to 2 instead of a fair 2 to 1. This is the casino’s main advantage.
To combat this, the player has options that allow him to have a flexible strategy. He can fold if he wants, spread his arms, leave, etc. But more importantly, he knows the dealer’s card which will force him to make those decisions (as opposed to simply hitting or standing) with that card in mind, and that’s where the math comes in.
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In the 1950s, a group of mathematicians led by Roger Baldwin developed a basic strategy. It is based on optimizing the decisions of the players in such a way that the action they choose is the most profitable in terms of all the actions they can take. This gives us the highest possible return in the long run at the beginning.
And it achieves this by developing a probabilistic model that calculates, based on the player’s hand, given the dealer’s card, when the most optimal decision to make is to stand, hit, double, split, or bet for insurance, both for hard inflexible hands and soft hands (where there is an ace that can change the amount).
Mathematically speaking, with a recursive function with defined end values and a table of all probabilities for the dealer’s sum, the maximum between the expected payoff, if it holds, and
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