Las Matemáticas del Blackjack

Las Matemáticas del Blackjack


What do you know about the Blackjack game? You’re curious but you don’t dare to play? Or have you played, but you have more losses than gains? You think you have a strategy for defeating the house? In either case, I recommend you to be attentive, in this video I will analyze Blackjack at a mathematical level so that you know before playing, what your real possibilities are so… let’s start with the logo and go with the video! Well, here I am again at the casino but this time to show you another game that’s Blackjack Although first I have to teach the rules and how people play so… let’s go with the video! Blackjack is a French deck game for up to 7 players where the players face the house The goal is to get 21 with our cards or a higher value than the bank, without exceeding the number If we exceed 21 or add a lower value than the dealer we’ll automatically lose our bet Once the bets are placed two visible cards are dealt to each player and only one for the dealer Players must say based on their cards and the croupier’s if they want to hit, stand, double down, or split their hand All cards are worth the numerical value they have, i.e. from 2 to 10, except for the figures that are worth 10, and the Ace that can be worth 1 or 11 to our interest Payments are made on a par, and we can add 21 with more than two cards but will only be considered blackjack when done with 2 and will be above any 21 added with more cards The house can only hit or stand and only hits when its hand is below 17 so if it reaches or exceeds that value will stand automatically in case the dealer goes over 21 the players who are still at the table will win their bets and in case of a tie the player will regain his bet You can play blackjack with one or more decks being 6 the standard in the European version The American version has some different rules that allow you to modify if the dealer hits or how players can split their hands In addition he deals himself a second card covered and you can consult it in case his first card give him the chance to do blackjack If effectively he gets 21, the players they’ll lose their bet before they even play Now that we’ve seen the rules, let’s take a closer look at what actions we can and what the payments are, so… Let’s go with it! In the first place, we can hit, while our sum does not exceed 21 depending on the card they give us we’ll be closer to 21, or instead We’ll lose the bet if we get over it. To double down we need a hand which obligatorily sums 9, 10 or 11 and you can only do it at the beginning of the round If you double your bet, you can only receive one more card If our cards have the same value, we can split each card into different hands to play them at the same time, but for that it is obligatory to add also a bet as the initial You must also know that within a split hand, even if we sum 21 with the next card is not considered blackjack If the card uncovered by the dealer is an Ace we can bet to insurance, if we feel that the house will get Blackjack with the next card If we’ve bet and finally gets it, we’ll be rewarded And finally surrender, which is only possible in the American Blackjack and if the player surrenders, he will lose only half the bet The payments are: 1 chip for each chip bet if we beat the house 3 chips, for every 2 chips bet if we beat the dealer with blackjack (as long as there is no tie) and finally, 2 chips for each chip bet if we win an insurance bet Now that you know what actions and awards we can take we’re going to investigate how the cards are distributed to understand a little better how the game works So… don’t miss it! To analyze Blackjack at a mathematical level we must know that a French Deck has 52 cards 4 of them are Aces, 16 of them are 10 or face card and the remaining 32 are numbers from 2 to 9 so we can draw some conclusions Approximately one third of the cards, have the value 10
Which takes a fundamental role After that, there are only 4 cards that are Ace so if we want to get a blackjack We’re gonna need one of them but… what are the odds? The probability of getting Blackjack would be the ways we have to get 21 with 2 cards of all possible combinations to pull 2 cards out of the deck Or put another way, favorable cases between possible cases To get 21 with 2 cards we need an Ace and a 10. and in the deck there are 4 Aces and 16 tens (between face cards and 10 value cards) On the other hand, all combinations of 2 cards are: combinations of 52 elements taken 2 at a time where the order doesn’t matter, and they can’t be repeated. So the final probability will be: 4X16, divided between the combinations of 52 elements taken 2 at a time which is equal to 64 between 1326 Or in other words, a 4.8% chance of getting blackjack In case of playing with several decks the proportion of each figure or number is still maintained since it’s the same 4 aces in 52 cards, than 8 in 104 but the odds of getting a particular result vary slightly since it influences less to remove a card in a deck of 104 than in a deck of 52 So each game with different decks has its own probabilistic study Now you’ve got an idea of how the cards are distributed but… What do you think about asking for one more card? Do you think it is easy to lose at first if we ask for another card? Let’s see it! To calculate the probability of going over 21 We must to calculate the probability of going over for any hand and for that, we need to go hand in hand Let us imagine that our hand value is for example 12 value when we hit, we are faced with all these cases As we see, initially there are 4 cases of 13 where we exceed 21 This is approximately a 30% chance This means that with a 12 value only 30% of the time we hit, we will exceed 21 If we calculate the same for all possible hands and accurately we will obtain the following table It’s simply telling us what the probability of go over, with the hand we have As you can see, from 13, both us and the house we’ll surpass the most of the time Also note that if our sum is 11 or less we can never surpass 21 with one more card That’s why his probability is 0 As we just saw, if we don’t want to lose most of the time we must hit, as long as our hand sum less than 14 the question is… If we put a limit on 14, do we have a chance of winning? If we consider that the dealer always hits for a card, until sum 17 or more Stand with a value less than 17, only makes you a winner, when the dealer exceeds or to put it another way, if the dealer doesn’t go over 21 His hand will always have a value equal to or greater than 17 so the players who stood with a hand less than 17 will lose their bet Well, according to the mathematical study carried out The probability that the dealer exceeds 21 depending on the card he has are the following We see that for a high card the probability of go over is quite low being the Ace the card that offers the most advantages to the house for a low card the probability is higher but note that in no case does it exceed 50% If we average all of them, we conclude that the croupier surpasses 21 in 28.35% of the cases, which means that in the remaining 71.65% of the times you stand with less than 17 You’re gonna lose your bet
As we have just seen, it’s useless to set a limit below the house limit since we will lose in most cases but thanks to this we’ve discovered that the dealer’s up card, offers us valuable information to make decisions After this we have the following questions: Why is the house limit 17? and second, what if I copy the house strategy and I also stand from 17? Will I have the same options as the dealer? or… maybe not? Let’s see it! If you look, the dealer always waits the player finishes his play before proceeding to play him that means that if the player goes over 21, he loses his bet regardless of what the dealer does! So if the dealer also goes over 21 it is the player who loses anyway since he had been eliminated before That makes the house wins in a case, which is really a tie and this happens 7.9% of the time you play That’s about an 8% advantage over the player which is finally reduced to 5.6%. if we consider that blackjacks are payed 3 to 2, instead of the equitable 2 to 1 That’s the casino’s main advantage! To fight this, the player has options that allow him to have a flexible strategy He can double down when he’s interested, split hands, surrender, etc. But most importantly, He knows a dealer’s card that will make him make those decisions
(differents from hit or stand only) considering this card And this, is where math comes in, once again In the 50s, a group of mathematicians led by Roger Baldwin developed the Basic Strategy This is based on optimizing player decisions
in such a way that the action he chooses It is the most optimal in terms of profit from among all the actions he can take This gives us at first, the maximum possible long-term gain and is achieved by developing a probabilistic model which calculates based on the player’s hand conditioned by the dealer’s card when it is more optimal to make the decision of stand, hit, double down, split or bet on insurance either for hard hands that are inflexible
as for soft hands (where there is an Ace and can change the sum) In mathematical language we can say that through a recursive function with definite final values and a table with all the probabilities for the sum of the dealer the maximum is calculated between the expected profit if stand and the expected profit considering all the possible cards he can receive when hits This produces a table of results, which assure us mathematically what is the most convenient action making our strategy optimal, whatever it is the combination of cards between the player and the dealer So if you want to play Blackjack well
I recommend you download this table As a detail, notice that in hard hands from 17, it gives us instructions to stand That means that regardless of the card have the dealer, math tell us that the expected profit is always higher if we stand That answers the question of why the house has 17 as a limit If we understand that the house is also a player! Look also, you should never bet on insurance This is because although the dealer has an Ace mostly he won’t get blackjack Well, with this strategy we will convert the previous 5.6% in a percentage lower than 1% for the house which will oscillate depending on the rules of the table and on the decks used We have to understand that although it is an optimal strategy does not imply that it is necessarily a winning strategy because the balance of our advantage despite being small, it is still negative This is mainly due to the dealer’s high cards diminish our advantage more than low cards So, even if we make the best decision we will not necessarily win, although in this way we will play in the best possible way In American Blackjack, the percentages are slightly different but in no case do they exceed 1% and only when you apply a Basic Strategy adapted to these rules remember, each modality is different and needs a different Basic Strategy If the best way to play it only minimizes our losses… Is there a real way to beat Blackjack? To answer this question, we must understand what hypotheses have been made to solve this problem and that is that the Basic Strategy was developed
from the beginning, in such a way that the odds of the cards are invariable throughout the game That means they’re always the same. which isn’t quite real since we know that the odds are changing while the cards are being dealt but … what if I told you, we have a system to have control of the cards that are appearing and in this way be able to anticipate winning hands? Would you be interested? Well, let’s see it! This counting system is known as Hi-Lo and wrote it for the first time in 1962 a mathematician former IBM employee in the book “Beat the Dealer.” The idea is to control the high cards of the deck which are the only ones who can conform blackjack so that if we know how many high cards have appeared, we know if there are many or a few to appear so we can know what odds we have to get blackjack in real-time with the additional advantage that this implies To know this advantage, we need to have control over the cards that are appearing something that seems to be quite complicated
if for that we need to memorize all the cards that have appeared Fortunately, mathematics offers us a better solution The Hi-Lo system associates 3 different values ​​to the cards -1,0 and 1 -1 for high cards +1 for low cards and the remaining ones acquire a null value or 0 Our task is then, to keep track of all the cards that have been dealt including the one from the dealer so if our account is negative it’s because we will have counted more high cards and there will be less to appear Something that doesn’t interest us Instead when our account is positive It’s because we will have counted more low cards and we will know for sure that there are more high cards to appear It is here that we can take advantage Our goal with this system is to accumulate in that account the largest possible positive number which offers us the highest probability of get blackjack If we get it, we can say that the table is hot and we can modify the basic strategy in our interest, for example to double down or bet on insurance How blackjack pays 3 to 2 and tie cases are unlikely we have for the first time a game system with a expected gain higher than the house And all thanks to mathematics! Notice, that the low cards balance the high cards which implies that the total account once all the cards have been dealt has to be 0 obligatorily In addition we also see that there are 3 cards that are not considered either high or low This is because the system You have to balance 5 high, with 5 low But in the French deck there are more numbers than faces, aces and tens So there are 3 leftover numbers that are grouped into a neutral value that does not interfere with the sum In the case of playing with several decks we must bear in mind that it is not the same a sum with 4 pending decks that the same sum with one That’s why you have to divide the current sum by the number of remaining decks and this will be the real account There are other counting methods a little more efficient, but assign more values ​​and are a little bit more complicated to execute But if you understand Hi-Lo, you will understand any other Although we can have an advantage with this system everything happens after thousands and thousands of hands and gives us an advantage of tenths of a percentage so you will need a very large wallet and a long time Unfortunately the casinos have introduced measures to combat the card counters The first is to increase the number of cards since the more decks the game uses the more complicated you will have it to win by counting think that the fact that exist more pending decks makes it harder to accumulate only the best cards at the end, that with a single deck That makes the account variance what are you doing, be smaller and consequently have less chance The second measure that is currently applied are the mixers or automatic machines to mix the cards, before the game is over that force the accountant to restart the account that had accumulated and therefore to lose all the advantage In addition some casinos such as “Las Vegas” they pay 6 to 5 blackjack which triples the house advantage over the player And even in some if they suspect that a player is an accountant they can afford to modify the house bet The count is legal, but if they discover you you’ll be expelled from the casino Casinos know the profile of an accountant in such a way that it will not be very complicated discover you, if you start to make money From the point of view of numbers there is a mathematical possibility of winning blackjack That’s why many people have earned money But it is not easy to execute neither fast nor much less cheap so it probably isn’t within your reach In any case if you are going to play I recommend you learn and adapt first the Basic Strategy to improve your chances Do not bet on tables with prizes below 3 to 2 Play at tables with the number of decks smallest possible And check the table rules very well And if you are going to count cards … study how they mix and test your strategy first before investing a lot of money Remember that: The more decks, the more the house wins the worse the rules, the more the house wins and the less strategy or knowledge
the more sure and faster wins the house so … does the house always win? does not always win, but it is clear that if it does not will change what is necessary to do it remember that this is a business If you liked the video give me a like or subscribe to the channel and if you have any questions about another gambling game leave me a comment and I will answer and that’s all so, thanks for your time and see you in the next video, greetings!