Welcome 🙂 Okay this is a bit of a special Mathologer today. A number of you have requested that I do something on blackjack and card counting so here we go–how to gamble yourself to fame and fortune. I am being assisted today by fellow mathematician, longtime colleague and part-time gambler Marty Ross who is really good at this stuff and who has offered to share some of the mathematical secrets to coming out on top in gambling games like blackjack. Okay so let’s begin with a couple of puzzles. For the first puzzle suppose you’re looking to bet on roulette. The roulette wheel is numbered from 0 to 37 with 18 red numbers, 18 black numbers and the green 0. So the chances of red coming up is just under 50/50.

Now let’s suppose you’ve been watching the roulette wheel and of the last 100 spins red has come up 60 times. What should you bet will come up next: red, black, doesn’t matter? Sounds too easy? Well this probably comes as a surprise but most people get this one wrong. We’ll give the answer in a little while. Our second puzzle actually arises in practice–a standard way that casinos and gambling sites sucker people into betting. For this puzzle you’re given a $10 free bet coupon. You can use the coupon to place a bet on any standard casino game: roulette, blackjack, craps, and so on.

If your bet wins then you receive the normal winnings. For example, let’s say you bet red on roulette. If red comes up you win $10, of course. Win or lose, the casino takes the coupon. Now here’s the question: what is the value of this coupon? In other words, what should or would you be willing to pay for such a coupon? We leave that one for you to fight over in the comments. But we’ll give you a hint: whatever you think the obvious answer is you’re definitely wrong 🙂 Now on with making our fortune. Famously the mathematician Blaise Pascal sorted out the basics of probability in order to answer some tricky gambling questions. When not dropping rocks Galileo also dabbled in these ideas.So if we roll a standard die, then there’s a one in six chance that five will come up, on a roulette wheel there is a 1 in 37 chance that 13 comes up, the usual stuff. And then comes in the money. What really matters to a gambler is not only the odds of winning but of course also how much they get paid if they win. right? And that is the idea of expectation, the expected fraction of the gamblers bet he expects to win or lose.

As an example, suppose we bet a dollar on red on roulette. We have an 18 in 37 chance of red in which case we win $1. There’s also a 19 and 37 chance of losing $1. And so, if we keep betting $1 on red, on average we expect a loss of 18/37 – 19/37 which is – 1/37th of $1, or -0.03 dollars. What this tells us is that in the long run we expect to have lost about 3% of whatever we’ve bet. 37 spins and we expect to have lost about one dollar. 370 spins and we’ve lost about $10 and so on. Of course, dumb luck can mean that the actual amount we might win or lose may vary dramatically. Again, in maths we express all this by saying that the expectation of betting on red is – 1/37th or minus 3%. As another example, what if you bet that the number 13 comes up? If 13 comes up we win $35 and there’s a 1in 37 chance of that. There’s also a 36 and 37 chance of losing your dollar and so our expectation comes to 35/37 – 36/37 or -1/37 which as in the first roulette game that we considered is equal to minus 1/37. In fact, no matter what you bet on roulette, the expectation will always be – 1/37 give or take some casino variation. Expectation can vary dramatically on gambling games, from close to 0% on some casino games down to -40% or so on some lotteries.

But, unsurprisingly, the expectation is pretty much guaranteed to be less than zero and minus means losing. So far so really really 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 such schemes is the so called martingale. This betting scheme works like this: as before let’s bet on red in roulette and let’s start by betting $1. If red comes up you win $1 and you repeat your $1 bet. If red does not come up you lose your dollar. To make up for your loss you play again but this time with a doubled wager of $2. If red comes up you win $2 which together with the $1 loss in the previous game amounts an overall win of 2 minus 1 is equals $1. So you’ve won, so you go back to betting just $1. On the other hand, if red does not come up you lose your $2 which then adds up to a total loss of 2 plus 1 is 3 dollars. You’ve only lost so far so you play again, but this time with a doubled wager of $4. If red comes up you win $4 which together with the $3 loss so far means that overall you’ve won $1. You’ve won and so you revert to betting just $1.

On the other hand, if red does not come up you lose your $4 which then adds up to a total loss of 4 plus 3 equals 7 dollars. So far you’ve only lost so you play again but this time with a doubled wager of $8, etc. So basically you keep doubling your bet until your bad luck runs out at which time you start from the beginning by betting $1 next then keep doubling your bet again until you win, and so on. As long as you stop playing after some win, this betting strategy seems to guarantee you always coming out on top overall. There are many such betting schemes the d’Alembert the reverse Labouchere. Apparently these schemes work much better if they have fancy French names, believe it or not. But do bet variation schemes work? Probability questions like this one can be tricky, depending in a subtle way on our assumptions. The martingale, for example, obviously works if you happen to have infinitely 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 we expect to happen? Well, suppose we make a sequence of bets with the same expectation for each bet, as in the setup we just looked at. Then the total amount we expect to win or lose is easy to calculate.

It’s just E times that positive number there and if E is negative then uhoh no luck. That brings us to the fundamental and very depressing theorem of gambling. The theorem says that if the expectation is negative for every individual bet then no bet variation can make the expectation positive overall. Damn ! 🙂 Okay, so we’re not going to get rich unless we somehow find a game with positive expectation. For the moment, let’s just assume that such a game exists. How well then can we do? Suppose we’re betting on a casino game for which the chances of winning are 2/3 and therefore a chances of losing are 1/3. Let’s also assume that just like in betting on red in roulette you win or lose whatever amount you bet. Then the expectation for this game is actually positive. To be precise it’s a whopping 33%. Now such a huge positive expectation in the casino game is clearly a fantasy. But bear with us. Ok, suppose we start with $100. What are the chances of doubling our money to $200? Well, obviously, if we just plunk it all down in one big bet of $100 then the chances of doubling are, well, 2/3, of course. This may come as a surprise but we can actually improve our chances if we bet $50 at a time and we play until we are either bankrupt or we have doubled our money. Let’s do the maths. If we place bets of fifty dollars, after one bet, win or lose, we either have 150 or 50 dollars. And after two bets we have $0, $100 or $200. Now, reading off 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 to be back to where we started from with $100 is, well, 2/3 times 1/3 plus 1/3 times 2/3 which happens to also be 4/9. But if we’re back at $100 we can keep on playing until eventually we have doubled our money or are bankrupt. It can actually take it while before this is sorted out, right? Now if D are the chances of eventually doubling our money in this way, then D is equal to what? Well, 4/9 the probability of having doubled our money after two bets plus the second 4/9 the probability of being back where we started from times the probability to be 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 quite a nifty calculation when you think about it.

Anyway, now we just have to solve for D and this gives that D is equal to 4/5 which is 80%. And this is definitely a lot better than 66% that going for just one bet of $100 guaranteed. Repeating the trick, we can consider betting 25 dollars at a time. This results in an about 94% chance of doubling our money. In fact, by making the bet size smaller and smaller we can push the probability of us eventually doubling our money to as close to certainty as we wish and once we’ve doubled our money, why not keep on playing to quadruple, octuple, etc. our money. And since we can push the probability of doubling our money as close to certainty as we like, the same is then also true for of those more ambitious goals. 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 second very encouraging theorem of gambling. So here we go. If the expectation is positive, then we can win as much as like, with as little risk as we like, by betting small enough for long enough. And so, finally, a bit of very good news, right? Alright, so all that’s holding us back from fame and fortune is finding a game of positive expectation. For that, of course, we again turn 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 which means that the true odds aren’t exactly what the simple mathematics predicts. Is this sufficient to get an edge on 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 and occasionally succeeded in beating a casino in this way and such attempts continue to this day. And with that in mind, we’ll now answer our roulette puzzle from the start. So if 60 of the last 100 spins have turned up red, then you should most definitely bet on red. Of course, feel free to disagree vehemently in the comments. Ok so finally on to making our fortune at blackjack, a possibility made famous in the Kevin Spacey movie 21. Well Kevin’s out of favour, now so should watch The last casino instead, it’s a much better movie anyway. For this video we don’t really have to worry too much about the rules of blackjack, so here’s just a rough sketch. Now blackjack is played with a standard deck of 52 cards or nowadays a number of such decks. The goal is to get as close to 21 without going over. All face cards count as ten, the aces count as 1 or 11 the player can actually choose whichever works better for them.

In blackjack you’re playing against the dealer. You’re initially dealt two cards and the dealer just one, all face-up for everybody to see. You go first. You can ask for more cards one at a time until you either bust which means you go over 21 in which case you lose immediately or you stop before this happens. Then it’s the dealer’s turn who will deal herself cards like a robot until she hits 17 or above and then stops. The person closest 21 without having gone bust wins. The casino’s edge comes from you the player having to go first knowing only the dealer’s first card. So if you bust by going over 21 then you lose immediately even if the dealer later busts as well. There are however some compensating factors that favor the player including the ability to make decisions such as when to stop receiving cards and whether to “split” or to “double”. We won’t go on to this. Actually the ability to make decisions only favors the player if they know what they’re doing which is actually hardly ever the case 🙂 The fundamentals of optimizing blackjack play involve knowing what decisions to make given any total of your cards and whatever the dealer’s card and this is known as “basic strategy”and was actually first figured out in the 1950s by some army guys playing with their new electronic calculators. The basic strategy can be summarized in a table which all expert players know by heart. Here’s a simplified version. Let’s use it. At the moment our cards add up to, well, 10 for the queen plus 5, that’s 15, so look up 15 on the left side. The dealer has 8 and so the basic strategy tells us that we should “hit” which means ask for another card. Let’s do that. Now we’ve got 19 and this means that the basic strategy tells us to stand or stop which of course makes total sense at this point in time. Figuring out the basic strategy just involves a lot of easy probability tree diagrams and stuff like that. Casino rules can differ which then changes the basic strategy slightly as well as the resulting expectation but in a not too nasty casino the expectation, given optimal play this way, might be about -0.5%.

Close but no banana. Of course plenty of people do worse than that. Casinos play their cards close to their chests but it seems that on average the casinos make well over 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 counting comes in. Card counting arose in the early sixties, courtesy of mathematician Edward Thorp and the fundamental idea is very easy. Basic strategy assumes that any card has an equal likelihood of appearing next. Well it’s a fairly natural assumption to make if there’s NO other information to be had but of course there IS other information to be had as cards get dealt the probabilities change. In general, high cards are better for the player and low cards are worse. Then, as the cards are dealt out, the expectation changes and the expectation will be positive if sufficiently many low cards are dealt. That sounds like a lot of information to keep track of but counting simplifies it all down to keeping track of just one number called the running count.

Every time the cards are shuffled the running count resets to 0. After the shuffle whenever you see a low card you add one to the running count. Whenever you see a high card you subtract one. Otherwise you don’t do anything. The running count indicates how many extra high cards there are among the cards left to be dealt. Keeping track of the the running count may seem tricky to do in a casino with all the cards zipping around on the table but it’s actually pretty easy watching a blackjack table for about an hour most people can keep track of the running count pretty accurately. There are also plenty of apps around like that one there 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 fast enough to keep track of the running count just now, over there. I showed this one to Marty cold and he just had it straight 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 remaining. For example five extra high cards matter much less if they’re within three decks left to be played than if there’s only one deck left to be played. To account for this we simply take the running count and divide by the number of decks left to be dealt.

This number is called the true count and here’s the surprisingly simple formula that relates the true count to the expectation at the given point of the game and this formula contains some really good news. A true count of two or greater means that our expectation is positive, right two minus one is positive. A true count of plus ten which can easily happen just before the shuffle 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 larger bets when the true count is higher. The bad news is that betting in such a manner involves a lot of boring waiting around followed by frantic and really really suspicious betting perhaps hundreds of dollars on a few brief hands. How well does it work? Well these days a typical betting scheme going up to say a maximum bet of 200 dollars might result in an average of about 15 dollars an hour. Wow, hmmm not what I would call a great hourly pay. And it gets worse, the result in any given hour can differ massively. 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 his run-ins with casinos. So unless you’re part of a well drilled team of counters and players or you’re really good at disguises there’s a fair chance you get to meet some burly casino employees within a few short hours. Well we did say blackjack is a way to win a SMALL 🙂 fortune. Good luck happy gambling and that’s all for today … Except we’ve all heard that back in the 70s there were lots of people making millions of dollars playing blackjack in the casinos. So what has changed? Why can’t we make millions of dollars these days. (Marty) well the casinos have gotten a lot more careful and a lot smarter: they use more decks which means the running count matters less, the true count is slower to get going, they use automatic shuffling machines, they really are on the lookout for suspicious betting. So unless you’re incredibly good at disguising yourself, incredibly good at team playing, it’s pretty much dead. (Burkard) It’s dead, that’s sad but what about other games? There’s online gambling now so are there other ways to make money with gambling these days. Absolutely yeah the casino is always looking to sucker more people into betting and suckering old people into betting more more, so there’s always promotions, there’s new games, new rules, some are knowingly have expectation which is positive and they just watch out, others the casino makes mistakes or online betting sites make mistakes. So you do a little expectation calculation and often not always but often you can find a little edge and enough of these little edges and you can make a nice little profit on the side and definitely there’s some people who just computerized everything, calculate to the nth degree and there’s some secret people I’m sure who are doing very very well. All right. Well that’s a perfect lead-in to our next video, at some point. Anyway thanks Marty for coming today. Thank you and we’ll have you again soon.

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