Win a SMALL fortune with counting cards-the math of blackjack & Co.May 01, 2020
Welcome :) Well, this is a little special Mathologer today. Several of you have asked me to do something about
counting, so here we go: how to bet for fame and
fortune. Today I am being assisted by fellow
mathematician, long-time colleague and part-time player Marty Ross, who is really good at this stuff and has offered to share some of the
mathematical secrets to winning at gambling games like
blackjack. . Okay, let's start with a couple of riddles. For the first puzzle, suppose you want to bet on roulette. The roulette wheel is numbered from 0 to 37 with 18 red numbers, 18 black numbers and a green 0.
Therefore, the chances of red coming up are a little less than 50/50. Now suppose you have been watching the roulette wheel and out of the last 100 spins, red has come up 60 times. What should you bet will appear next: red, black, it doesn't matter? Sounds too easy? Well, this is probably a surprise, but most people are wrong. We will give the answer in a bit. Our second puzzle actually arises in practice: a standard way casinos and betting sites trick people into betting. For this puzzle, you receive a $10 free bet coupon. You can use the coupon to bet on any standard casino game: roulette, blackjack, craps, etc.
If your bet wins, then you receive the normal winnings. For example, suppose you bet on red in roulette. If it comes up red, you win $10, of course. Win or lose, the casino takes 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 think, the obvious answer is that you're definitely wrong :) Now let's make our
fortune. The famous mathematician Blaise Pascal worked out the basics of probability to answer some tricky questions about gambling.
When he wasn't throwing stones, Galileo also ventured into these ideas. So if we roll a standard dice, then there is a one in six chance that it will come up with five, on a roulette wheel there is a one in 37 chance that it will come up with 13, which is normal. And then comes the money. What really matters to a player is not just the odds of winning but of course also how much they get paid if they win. right? And that's the idea of expectation, the expected fraction of the player's bet that he expects to win or lose.
As an example, suppose we bet a dollar on red at roulette. We have an 18 in 37 chance of it coming up 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 run we expect to have lost around 3% of what we have bet. 37 spins and we expect to have lost about a dollar. 370 spins and we have lost about $10 and so on. Of course, dumb luck can mean that the actual amount we stand to win or lose can vary drastically.
Again, in mathematics we express all of this by saying that the expectation of betting on red is -1/37 or minus 3%. As another example, what happens if you bet the number 13 comes up? If 13 rolls, we win $35 and there is a 1 in 37 chance of that. There is also a 36 and a 37 probability of losing your dollar, so our expectation is 35/37 - 36/37 or -1/37 which, as in the first game of roulette we considered, equals 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 drastically in gambling games, from close to 0% in some casino games to -40% or so in some lotteries.
But, unsurprisingly, 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, one popular trick is to vary the size of your bet depending on whether you win or lose. The most famous of such schemes is the so-called martingale. This betting scheme works like this: as before, let's bet on red in roulette and start by betting $1. If it comes up red, you win $1 and repeat your $1 bet. If it doesn't come up red, you lose your dollar. To make up for his loss, he plays again, but this time with a doubled bet of $2.
If it comes up red, you win $2 which, together with the loss of $1 in the previous game, adds up to a total win of 2 minus 1 which equals $1. So you've won, so you bet again just $1. On the other hand, if red doesn't come up, you lose your $2, which then adds up to a total loss of 2 plus 1 is $3. 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 win $4, which together with the $3 you've lost so far means that overall you've won $1. You have won, so you bet again only $1.
On the other hand, if red doesn't come up, you lose your $4, which adds up to a total loss of 4 plus 3 equals $7. So far he has only lost, so he plays again, but this time with a doubled bet of $8, etc. You basically keep doubling your bet until your bad luck runs out, at which point you start all over by betting $1 and then continue. doubling 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 come out on top overall. There are many such betting schemes, the d'Alembert to the reverse Labouchere.
Apparently these schemes work much better if they have fancy French names, believe it or not. But do variance betting schemes work? Probability questions like this can be tricky, since they depend 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 gambling? And of course, whatever you do, you can always get lucky, but with a finite amount of money in your pocket, what can you 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 you're out of luck. That 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 change in the bet can make the expectation positive overall. Dammit ! :) Well, then 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 then?
Suppose we are betting on a casino game where the chances of winning are 2/3 and therefore the chances of losing are 1/3. Let's also assume that, just like betting on red in roulette, you win or lose the amount you bet. So the expectation for this game is really positive. To be precise, that's a whopping 33%. Now such a large 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 roll it all into one big bet of $100, then the chances of doubling down are, well, 2/3 of course.
This may come as a surprise, but we can actually improve our chances by betting $50 at a time and playing until we're 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 where we started with $100 is, well, 2/3 times 1/3 plus 1/3 times 2/3 which is also 4/9.
But if we get back to $100, we can keep playing until we've finally doubled our money or gone bankrupt. It may actually take a while before this is resolved, right? Now if D is the chances that we will eventually double our money in this way, then D equals what? Well, 4/9 chance of having doubled our money after two bets plus the second 4/9 chance of being back where we started because of the chance of being able to double from this point on. And what is that? Well, we're back to $100. So the probability is D again. It's actually a pretty nifty calculation when 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 66% of just betting for $100 guaranteed. Repeating the trick, we can consider betting $25 at a time. This results in about a 94% probability of doubling our money. In fact, by making the bet size
smaller, we can increase the probability that we will eventually double our money as close to certainty as we want and once we have doubled our money why not keep playing for quadruple, eightfold, etc.? . Our money. And since we can push the probability of doubling our money as close to certainty as we want, the same is true for 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 that we play is positive, as in the game that was played. The surprising conclusion of all this is our very encouraging second theorem of gambling. So here we go. If the expectation is positive, 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, some very good news, right? Alright, all that's holding us back from fame and fortune is finding a positive expectation set.
For that, of course, we turn again to the game of roulette. .. Just kidding :) and we'll get back to blackjack in a minute. But there are many approaches to gambling and one factor to keep in mind is that games like roulette are mechanical, meaning the actual odds are not exactly what simple math predicts. Is this enough to get 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 to beat a casino in this way, and on occasion have succeeded, and those attempts continue to this day.
And with that in mind, we will now answer our roulette puzzle from the beginning. So if 60 out of the last 100 spins have been 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 possibility made famous in the Kevin Spacey movie 21. Well Kevin is out of style, 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 blackjack rules, so here's just a rough outline. Blackjack is now played with a standard deck of 52
cardsor, nowadays, with several such decks.
The goal is to get as close to 21 as possible without going over. All face
cardscount as ten, aces count as either 1 or 11, the player can choose what works best for them. In blackjack you are playing against the dealer. Initially, he is dealt two cards and the dealer only one, all face up for all to see. You go first. He can draw more cards one at a time until he checks, which means he checks 21, in which case he loses immediately or stops before this happens. Then it's the dealer's turn, who will deal the cards like a robot until it reaches 17 or higher and then stops.
The closest person to 21 who hasn't passed wins. The casino advantage comes from the fact that you, the player, have to go first knowing only the dealer's first card. So if you go over 21, then you lose immediately, even if the dealer checks later as well. However, there are some compensating factors that go in the player's favor, including the ability to make decisions such as when to stop taking cards and whether to "split" or "double down". We will not continue with this. In reality, the ability to make decisions only favors the player if they know what they are doing, which in reality is almost never the case :) The fundamentals of blackjack game optimization involve knowing what decisions to make given the total of your cards and whatever the dealer's card is. and this is known as "basic strategy" and was actually first discovered in the 1950's by some military men tinkering with their new electronic calculators.
Basic strategy can be summarized in a table that all skilled 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 find 15 on the left side. The dealer has 8, so basic strategy tells us to "hit", which meansask for another card Let's do that. We now have 19 and this means that basic strategy tells us to stay or stop, which of course makes a lot of sense at this point. Figuring out basic strategy just involves a lot of easy probability tree diagrams and the like.
Casino rules may differ, which slightly changes the basic strategy, as well as the resulting expectation, but in a not-too-nasty casino, the expectation, given optimal play like this, might be about -0.5%. Close but no banana. Of course, many people do worse than that. Casinos play their cards carefully, 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 our fortune, we have to somehow get around that -0.5% and that's where card
countingcomes into play. Card counting emerged in the early 1960s courtesy of mathematician Edward Thorp, and the underlying idea is easy.
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 the odds 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. That sounds like a lot of information to keep track of, but counting makes it simple to keep track of a single number called a checking account.
Every time the cards are shuffled, the running count is reset to 0. After shuffling, whenever you see a low card, you add one to the running count. Every time you see a high card, you subtract one. Otherwise you do nothing. The running count indicates how many extra high cards remain among the cards left to be dealt. Keeping track of your checking account may seem difficult to do in a casino with all the cards spinning on the table, but it's actually quite easy to stare at a blackjack table for an hour or so, most people can keep track. of the current account quite accurately.
There are plenty of apps like that out there too 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 follow up on the checking account right now, there? I showed this one to cold Marty and he had it right away. However, 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 a lot less if they're in the three decks left to play than if there's only one deck left to play.
To account for this, we simply take the running count and divide it by the number of decks left to be dealt. This number is called the actual count and here is the surprisingly simple formula that relates the actual count to the expectation at the given point in the game and this formula contains very good news. A true count of two or more means that our expectation is positive, to the right two minus one is positive. An actual count of plus ten that can easily occur just before the mix means the expectation is 4.5%, which is pretty amazing.
So what does the card counter do? Well, ideally, she bets little or nothing when the true count is negative, makes small bets if the true count is slightly positive, and then makes bigger 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, maybe hundreds of dollars in a few short hands. How well does it work? Well, these days, a typical betting scheme that goes up to say a $200 max bet could result in an average of around $15 an hour. Wow, hmmm not what I would call a great pay per hour.
And what is worse, the result in a given hour can differ greatly. You can expect one standard deviation, a typical plus or minus, to be around $500. Of course, the way card counters bet makes them very easy to spot, and Marty has had his problems with casinos. So unless you're part of a well-trained team of accountants and players or you're really good at dressing up, there's a good chance you'll meet some burly casino employees in a few hours. Well, we said that blackjack is a way to win a LITTLE 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 winning millions of dollars playing blackjack in casinos. So what has changed? Why can't we make millions of dollars these days. (Marty) well, casinos have gotten 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 automated shuffling machines, they really keep an eye out for bets suspicious. So unless you're incredibly good at dressing up, incredibly good at team play, it's pretty much dead. (Burkard) It's dead, that's sad, but what about other games?
There is gambling online now, so there are other ways to make money gambling these days. Absolutely yes, the casino is always looking to trick more people into betting and tricking older people into betting more, so there are always promotions, there are new games, new rules, some knowingly have expectations which is a positive and just be careful, others the casino make mistakes or online gambling sites make mistakes. So you do a little expectation calculation 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, calculate to the nth degree and there are some secret people that I'm sure are doing very well.
Everything's fine. Well, that's a perfect introduction to our next video, at some point. Anyway, thanks Marty for coming out today. Thank you and we will have again soon.
If you have any copyright issue, please Contact