Win a SMALL fortune with counting cards-the math of blackjack & Co.

Welcome :) Well, this is a bit of a special Mathologer today. Several of you have asked me to do something about

blackjack

and card

counting

, so here we go: how to bet on fame and

fortune

. Helping me today is

math

ematician, long-time colleague and part-time player Marty Ross, who is really good at this topic and who has offered to share some of the

math

ematical secrets to getting to the top in games of chance like

blackjack

. . Okay, so let's start with a couple of riddles. For the first puzzle, let's say you want to bet on roulette. The roulette wheel is numbered from 0 to 37 with 18 red numbers, 18 black numbers and the green 0.

Therefore, the chances of red coming out are just under 50/50. Now let's say you've been watching the roulette wheel and in the last 100 spins red has appeared 60 times. What should you bet will appear next: red, black, doesn't matter? Does it sound too easy? Well, this probably comes as a surprise, but most people get it wrong. In a while we will give the answer. Our second puzzle actually arises in practice: a standard way casinos and betting sites trick people into betting. For this puzzle you will receive a $10 free bet coupon. You can use the coupon to place a bet on any standard casino game: roulette, blackjack, craps, etc.

If your bet wins, you will receive the normal winnings. For example, let's say you bet on red in roulette. If it comes out red, you'll win $10, of course. Win or lose, the casino accepts the coupon. Now here is the question: what is the value of this coupon? In other words, how much should or would you be willing to pay for that coupon? We leave that one for you to fight in the comments. But we'll give you a hint: whatever you believe, the obvious answer is that you are definitely wrong :) Now let's continue making our

fortune

. Famous mathematician Blaise Pascal solved the basics of probability to answer some complicated questions about gambling.

When he wasn't throwing stones, Galileo also dabbled in these ideas. So if we roll a standard dice then there is a one in six chance of getting five, on a roulette wheel there is a 1 in 37 chance of getting 13, the usual. And then comes the money. What really matters to a player is not only the odds of winning, but also, of course, how much he will be paid if he wins. good? And that is the idea of ​​expectation, the expected fraction of the player's bet that he expects to win or lose. As an example, let's say we bet a dollar on red in roulette.

We have an 18 in 37 chance of getting red, in which case we win $1. There is also a 19 and 37 chance of losing $1. And so, if we continue to bet $1 on red, on average we expect a loss of 18/37 - 19/37, which is - 1/37 of $1, or -0.03 dollars. What this tells us is that in the long term we expect to have lost around 3% of whatever we bet. 37 spins and we hope to have lost about a dollar. 370 spins and we've lost about $10 and so on. Of course, bad luck can mean that the actual amount we could win or lose can vary dramatically.

Again, in mathematics we express all this by saying that the expectation of betting on red is - 1/37 or minus 3%. As another example, what happens if you bet on the number 13 coming up? If it comes up 13, we win $35 and there is a 1 in 37 chance of making it. There is also a 36 and 37 chance of losing your dollar, so our expectation is 35/37 - 36/37 or -1/37 which, as in the first roulette game we considered, is equal to minus 1/37 . In fact, no matter what you bet on roulette, the expectation will always be: 1/37 plus or minus some casino variation. The expectation can vary dramatically in games of chance, from around 0% in some casino games to approximately -40% in some lotteries.

But, as expected, the expectation is guaranteed to be less than zero and less means losing. So far so very, very bad :) Hmm, what can we do about it? Well, a popular trick is to vary the size of your bet depending on whether you win or lose. The most famous of these systems is the so-called martingale. This betting scheme works like this: as before, let's bet on red on the roulette wheel and start by betting $1. If it comes up red, you win $1 and repeat your $1 bet. If red doesn't come out, you lose your dollar. To make up for your loss, you play again, but this time with a doubled bet of $2.

If it comes up red, you will win $2, which together with the $1 loss in the previous game equals an overall win of 2 minus 1, which equals $1. Then you've won, so you bet just $1 again. On the other hand, if red does not appear, you will lose your $2, which then adds up to a total loss of 2 plus 1, or 3 dollars. You've only lost so far so you play again, but this time with a doubled bet of $4. If it comes up red, you will win $4, which together with the $3 loss so far means that in total you have won $1. You have won and that is why you bet only $1 again.

On the other hand, if red does not appear, you will lose your $4, which then adds up to a total loss of 4 plus 3, which equals 7 dollars. So far you've only lost, so you play again, but this time with a doubled bet of $8, etc. Basically, you keep doubling your bet until you run out of bad luck, at which point you start from the beginning by betting $1 next and then continue. double your bet again until you win, and so on. As long as you stop playing after winning, this betting strategy seems to guarantee that you will always emerge victorious overall.

There are many such betting schemes, the d'Alembert and the reverse Labouchere. Apparently these schemes work much better if they have fancy French names, believe it or not. But do betting variation schemes work? Probability questions like this can be complicated, depending in subtle ways on our assumptions. The martingale, for example, obviously works if you have an infinite amount of dollars in your pocket. But then why bother betting? And of course, whatever you do you can always get lucky but with a finite amount of money in your pocket, what can we expect to happen? Well, suppose we make a sequence of bets with the same expectation for each bet, like in the setup we just saw.

So the total amount we expect to win or lose is easy to calculate. It's just E multiplied by that positive number there and if E is negative then no luck. This brings us to the fundamental and very depressing theorem of the game. The theorem says that if the expectation is negative for each individual bet, then no variation of the bet can make the expectation positive overall. Curse ! :) Okay, so we won't get rich unless we somehow find a game with positive expectations. For the moment, let's assume that such a game exists. How well can we do it then?

Suppose we are betting on a casino game in which the chances of winning are 2/3 and therefore the chances of losing are 1/3. Let's also assume that, just like when betting on red in roulette, you win or lose whatever amount you bet. So the expectations for this game are really positive. To be precise, it is a whopping 33%. Now, such a positive expectation in casino gambling is clearly a fantasy. But bear with us. Ok, let's say we start with $100. What are the chances of doubling our money to $200? Well, obviously, if we invest it all in a big bet of $100, then the chances of doubling it are, well, 2/3, of course.

This may seem like a surprise, but we can actually improve our chances by betting $50 at a time and playing until we are broke or have doubled our money. Let's do the math. If we make bets of fifty dollars, after a bet, whether we win or lose, we have 150 or 50 dollars. And after two bets we have $0, $100 or $200. Now, reading the tree, we see that at this point the probability of having doubled our money in the first two plays is 2/3 times 2/3, which is equal to 4/9. And similarly, the probability of getting back to square one with $100 is, well, 2/3 times 1/3 plus 1/3 times 2/3, which also happens to be 4/9.

But if we have $100 again, we can keep playing until we've finally doubled our money or we're bankrupt. In fact, it may take some time before this is fixed, right? Now if D is the chances of eventually doubling our money this way, then D equals what? Well, 4/9 the probability of having doubled our money after two bets plus the second one. 4/9 the probability of being back where we started multiplied by the probability of being able to double from this moment on. And what is that? Well, back to $100. So the probability is again D. It's actually a pretty clever calculation if you think about it.

Anyway, now we just have to solve for D and this gives D equals 4/5, which is 80%. And this is definitely much better than the 66% you get with a single guaranteed $100 bet. Repeating the trick, we can consider betting 25 dollars at a time. This results in a probability of around 94% of doubling our money. In fact, by making the bet size

small

er and

small

er, we can increase the probability that we will eventually double our money as close as we want and once we have doubled our money, why not keep playing to quadruple, octuple , etc.? . Our money. And since we can get the probability of doubling our money as close to certainty as we want, the same applies to those more ambitious goals as well.

Even better, the same turns out to be true no matter what probabilities we're dealing with. As long as the expectation of the game we play is positive, as in the game that was played. The very surprising conclusion to all this is our very encouraging second theorem of the game. So, here we go. If the expectation is positive, then we can win as much as we want, with as little risk as we want, by betting small enough for long enough. And so, finally, very good news, right? Okay, so the only thing stopping us from achieving fame and fortune is finding a set of positive expectations.

To do this, of course, we turn to the game of roulette again. ..Just kidding :) and we'll get back to blackjack in a minute. But there are many ways to play and one factor to keep in mind is that games like roulette are mechanical, meaning that the actual odds are not exactly what simple math predicts. Is this enough to gain an advantage in the game? Well, I won't go into that today, but in the references you can find some fascinating stories of people who have tried and occasionally succeeded in beating a casino this way and those attempts continue to this day.

And with that in mind, we'll now answer our roulette riddle from the beginning. So if 60 of the last 100 spins resulted in red, then you should definitely bet on red. Of course, feel free to vehemently disagree in the comments. Well, we're finally going to make our fortune at blackjack, a prospect made famous in the Kevin Spacey movie 21. Well Kevin is out of favor, now we should watch The last casino, it's a much better movie anyway. For this video we don't have to worry too much about the rules of blackjack, so here's just a rough outline. Blackjack is now played with a standard 52-card deck or, today, with several such decks.

The goal is to get as close to 21 as possible without going over. All face

cards

count as ten, aces count as 1 or 11 and the player can choose the one that best suits him. In blackjack you play against the dealer. Initially you are dealt two

cards

and the dealer only one, all face up for everyone to see. You go first. You can draw more cards one at a time until you go bust, meaning you go over 21, in which case you lose immediately or stop before this happens. Then it's the dealer's turn, who will deal cards like a robot until he reaches 17 or more and then stops.

The person who gets closest to 21 without having gone bankrupt wins. The casino advantage comes from the fact that you, the player, have to play first knowing only the dealer's first card. So if you bust when passing 21, you lose immediately even if the dealer also busts later. However, there are some countervailing factors that favor the player, including the ability to make decisions such as when to stop receiving cards and whether to "split" or "double down." We will not continue with this. In reality, the ability to make decisions only benefits the player if he knows what he is doing, which is almost never the case :) The fundamentals of optimizing the game of blackjack involve knowing what decisions to make given the total of his cards and any be the dealer's card. and this is known as "basic strategy" and was actually first discovered in the 1950s by some military men playing with their new electronic calculators.

The basic strategy can be summarized in a table that all expert players know by heart. Here is a simplified version. Let's use it. Right now our cards add up to, well, 10 for the queen plus 5, that's 15, so look for 15 on the left side. The dealer has 8 and that's whyBasic strategy tells us that we must "hit", which means to ask for another card. Let's do that. We are now 19 and this means that the basic strategy tells us to stop or stop, which of course makes a lot of sense at the moment. Figuring out basic strategy just involves a lot of simple probability tree diagrams and things like that.

Casino rules may differ, which then slightly changes the basic strategy as well as the resulting expectations, but in a not too nasty casino the expectation, given optimal play this way, could be around -0.5 %. Close but without banana. Of course, many people do worse than that. Casinos play their cards close to their chests, but it seems that on average casinos win more than 5% on blackjack, a clearly better rate of return for the casino than on roulette. Anyway, if we want to make a fortune, we have to somehow get around that -0.5% and that's where card

counting

comes into play.

Card counting emerged in the early 1960s, courtesy of mathematician Edward Thorp, and the fundamental idea is very simple. Basic strategy assumes that any card has an equal chance of appearing next. Well, it's a pretty natural assumption if there is NO other information available, but of course there IS other information available as the cards are dealt and the probabilities change. In general, high cards are better for the player and low cards are worse. Then, as the cards are dealt, the expectation changes and the expectation will be positive if enough low cards are dealt. It seems like a lot of information to keep track of, but counting simplifies everything and allows you to keep track of a single number called a running count.

Every time the cards are shuffled, the running count is reset to 0. After shuffling, every time you see a low card, you add one to the running count. Every time you see a high card, you subtract one. Otherwise you don't do anything. The running count indicates how many extra high cards remain among the cards left to be dealt. Keeping track of the count may seem complicated in a casino with all the cards spinning around on the table, but it's actually quite easy to watch a blackjack table for about an hour and most people can keep track of the count fairly well. precision.

There are also plenty of apps like that if you want to practice in the safety of your home or you can just get a plain old deck of cards. Now, were any of you quick enough to keep track of the running count just now, over there? I showed it to Marty and he had it right away. Anyway, what we really want to know is not the number of extra high cards left to be dealt, but the fraction of extra high cards left. For example, five extra high cards matter much less if they are within the three decks left to play than if there is only one deck left to play.

To account for this, we simply take the cumulative count and divide it by the number of decks left to deal. This number is called the true count and here is the surprisingly simple formula that relates the true count to the expectation at a given point in the game and this formula contains some really good news. A true count of two or more means our expectation is positive, two minus one is positive. A real plus ten count that can easily occur just before the mix means that the expectation is 4.5%, which is quite surprising. So what does the card counter do?

Well, ideally you should bet little or nothing when the true count is negative, make small bets if the true count is slightly positive, and then make larger bets when the true count is higher. The bad news is that betting this way involves a lot of boring waiting followed by frantic and really suspicious betting, perhaps hundreds of dollars in a few short hands. How well does it work? Well, today a typical betting scheme that goes so far as to say a maximum bet of $200 could result in an average of about $15 per hour. Wow, hmmm, not what I would call a great hourly wage.

And what's worse, the result at any given time can vary greatly. You can expect a standard deviation, a typical plus or minus, to be about $500. Of course, the way card counters bet makes them very easy to spot and Marty has had problems with the casinos. So unless you're part of a well-trained team of accountants and players or you're really good at disguising yourself, there's a good chance you'll meet some beefy casino employees within a few hours. Well, we said that blackjack is a way to win a SMALL :) fortune. Good luck, happy gaming and that's all for today...

Except we've all heard that in the 70's there were a lot of people who won millions of dollars playing blackjack in casinos. So what has changed? Why can't we earn millions of dollars today? (Marty) Well, the casinos have become a lot more careful and a lot smarter: they use more decks, which means the checking count matters less, the real count is slower, they use automatic shuffling machines, they really pay attention. for suspicious bets. So unless you're incredibly good at disguising, incredibly good at teamplaying, it's pretty much dead. (Burkard) It's dead, that's sad, but what about other games?

There is online gambling now, and there are also other ways to make money gambling these days. Absolutely yes, the casino is always looking to trick more people into betting and trick older people into betting more, so there are always promotions, there are new games, new rules, some know they have expectations which is positive and they are simply attentive, others the casino makes mistakes or the online betting sites make mistakes. So you do a little calculation of expectations and often not always but often you can find a little advantage and enough of these little advantages and you can make a little extra profit and there are definitely some people who just computerized everything, they calculate to the nth degree and I'm I'm sure there are some secret people who are doing very, very well.

Alright. Well, that's a perfect introduction to our next video, sometime. Anyway, thanks Marty for coming today. Thank you and we will have it again soon.