Gambling with the Martingale Strategy - Numberphile

Let's bet. We're creating a... let's call it a fictitious casino and we're playing a game with a 50/50 chance of winning. Okay, so we're doing roulette: red or black, so there's no zero, so it's really a 50/50 chance of red or black. What we are going to look at is something called the Martingale Strategy. Every time you lose a bet, you double the next bet; and therefore you will eventually win - is the idea. - (Brad: What do you mean) (double?) So if you bet a pound and lose, you bet again and you would bet two pounds; and then if you lose again, you lose four pounds;

If you lose, you gain eight pounds. Then you double the last bet you made. As long as you have virtually unlimited money, you will eventually win. A victory is: leaving our fictitious casino with more money than I entered. (You want to make a profit). - Exactly we are making profits. - (Should we say we are) (always betting on red?) (Or black or-?) - I think- I'm tempted to go black. So I write with a black pen. I think if given the choice, I would... I would go for the black over the red, I think. (Okay, so we bet on black.) - We bet on black.

And we will move forward; Do you think well when is the first time I lose? So if I lose on step k, I made my bet on step 1, and I bet a pound, and then I must have lost because we assume that we are going to continue until this number k. Then, in step 2, I bet two pounds. Then, on the third step (again assuming I lost on step 2), I double my bet, so it will now be four pounds. Step 4 will be double that, eight pounds, and this continues. - (If you win, your money doubles) (doubles?) - If I win, my money will double, yes, it's a 50/50 game where whatever you bet, exactly like roulette.

We don't have a zero, but otherwise the rules are the same. If I bet on black, which is what we're doing, if it comes out on black, I double my money. (Because that zero is how the casino insures itself). Zero, exactly. You know it this way. Zero is how casinos know and that's why they have zero, because there is never a 50/50 chance. But our mathematically fair casino gives us a true 50/50 chance. In step k we can calculate how much we would bet because there is a pattern here. So in step 1 I bet one pound, in step 2 two pounds and in step 3 four pounds.

So what's happening with our bets is that these are increasing powers of 2. So this is multiplied by 2, multiplied by 2, multiplied by 2. So the actual formula will be 2 raised to the power of k minus 1. So that we can verify this: so, in step 1, k is 1; 2 to the power of 0 is 1. Step 2 k is 2, 2 to the power of 1 is 2; 2 squared is 4, 2 cubed is 8, so in step k we just bet 2 to k minus 1 pound. If I then lose at step k, my total loss so far is the sum of all these bets; so 1 plus 2 plus 4 plus 8 up to 2 to the power of k minus 1.

We can write this as the sum of j equals 0 to k minus 1 of 2 power j. So 2 to the power of 0 is 1, then plus 2 to the power of 1 is 2 plus 2 squared is 4. This is a shorthand way of writing this sum. And this is what we call geometric series; We have a formula for the sum of a geometric series so we can substitute it. It is the first term, which is 1, multiplied by 1 minus the next term in the sequence. So it ends up at 2 to the power of k minus 1, so if we went to the next step it would be 2 to the power of k; and then we divide by 1 minus the common ratio between the terms.

So what we're multiplying in each step, we divide by 1 minus 2. So, you sort this and this is 2 to the power of k minus 1. So, this is the total amount of money I lost when I lose my bet on step k. So in the next step I am going to bet 2 on the k pounds; I hope to win and that will cover my losses with a pound to spare. So, after all this effort, after losing at some point, if I follow the Martingale

strategy

, I will eventually win - we hope so - before I run out of money, that's another matter, and I will win a pound.

So we're not exactly... like, well, it's a profit, but it's not like... we want more than a pound, right? As if you didn't really go to a casino to win a pound. (But if you scaled it to 10 pounds or to) (million pounds like) (like your- like your multiple) (you are guaranteed to win a million pounds!) Yes, to win a million pounds you have I had to repeat this

strategy

a million times and earn a pound every time. (Or just make that one million pound bet;) (a two million pound bet; four million) (pound bet, up to two) (up to minus 1 per million-) - We could do that too, yes. .

So we could also change the amount that we bet at the beginning and then that would be: the amount that you bet initially is the amount that you will win with this strategy. So this strategy will make you earn, through this process, we will earn a pound every time. Now, there are problems with this. There is a reason why this is not a foolproof gaming strategy. First of all, most games are not 50/50; If we are playing roulette, there is a zero that is green, so it is not red or black, so you don't really have a 50/50 chance.

And that really alters what's happening. - (In some casinos there are) (there are two zeros) Yes, zero and double zero, exactly yes, then they are even, those odds are even stingier. The other thing is that casinos have maximum betting limits. In most casinos you can't walk in and just drop a million pounds, as if they're going to say no, there's a maximum betting limit here. And even if you start very low and keep losing, this will grow very quickly. This is exponential growth that we know a lot about these days. And this will grow very quickly and will actually reach a million much faster than you think. - (So if it comes up red) (many times in a row - ?) - Yeah, you're going to hit the maximum betting limit in a real casino and then you're screwed, then you've lost like a million pounds. , you have no chance to get your money back.

Because that's the key; If this fails, you'll have lost 2 to the power k minus 1 pound, which is a lot of money if k is like 10 or something, so it can really go up. (Just for clarity, Tom, I) (I would imagine) (it doesn't really matter what you are) (placing the bets, red or black?) (As if you don't really have to bet) (on which color you lost? ) - No, no, not exactly. Each time, because there is an equal chance of each one appearing, you can switch. Maybe you could choose red, black, red, black, red, it doesn't matter because the probability will always be half and it will always be the same. (So ​​you don't have to bet on what) (you lost, you just have to, you just have to bet) (again). - You just have to bet again, yes exactly, you can bet on anything - in the 50/50 situation.

Another issue, no one in the world has unlimited money. It may seem like some people do it, but there is a number where the maximum amount any individual can bet. That's our bank balance, n pounds. If we ever need to bet more money than we have, we're screwed. Not only can we not earn our pound but we have literally lost everything we own. So we really don't want this to happen. (Do your previous winnings) (become part of that pot that you can dig into?) No, so the 100 pound limit here is... like that's my pot. So even if I start winning, maybe I've won 50 in a row and now theoretically I have 150, but what we're saying here is that 50 that I've won are in a separate pot.

That's like... that's in your money box, that goes to the bank, you can't touch it. We have said here that we are losing in step k and in the next step we would have to bet 2 raised to the power k. So if I have to bet 2 on the k pounds; If this ever gets bigger than my n pounds, it's game over. Sad face, that's all. - (You're broke) We all cry. So we cannot allow this to happen. Now what you can figure out is how does this tell you about k? Because what he remembers is the number of losses that were allowed.

So we can start plugging in some numbers here and thinking, well, if I come in with 100 pounds, how many losses am I allowed before I lose all of my hundred pounds? Just rearrange this slightly so that we can say that 2 to the power of k equals n, just to make things a little bit easier, so obviously if we go a little further than this, we're going to fail, so let's look at the limiting case of when' we are equal. So you take log to the base 2 of both sides and what it does is tell you that this is our value of k.

Now this is quite difficult to interpret because we don't actually have any numbers, so let's put some numbers in. So if I come up with n being 100 pounds. So if we start with 100 pounds, you're allowed seven losses, not too bad. But it may seem unlikely, I'm not going to lose seven times in a row, you'd be surprised how common it is to lose seven times in a row. Just like there are a lot of really interesting studies done on how bad we are as humans at thinking "oh, I'll never be that unlucky." It's very easy to lose seven times in a row. (Seven reds in a row and you're... are you out?) If we bet on black, yes.

And then if you go up to say a thousand pounds, you'll get about 10 losses. So even though you have multiplied your initial bet tenfold, you will only get three additional losses. And this kind of pattern continues, it starts to decrease. So if you had 10,000 pounds, then you would have 14 losses and if you even went as far as Brady's idea of ​​a million pounds, you would have 20 losses. You get double the losses from 10 to 20, going from a thousand to a million pounds. No matter how much your money increases, it begins to decrease drastically. (And yet this will only make you gain) (a pound!) - This is to gain a pound, yes, this is only to gain a pound.

So we have this relationship between the number of losses and the money we have invested. So what we can do is say, well, it's a 50/50 game, so what is the probability that you lose k times in a game? row? Well, if I lose once it's half, then if I lose again it's half; These are independent events, it doesn't matter what happened in the previous one. So it's literally just half to the power of k. So, if I have lost k times it is half to the power of k. This is actually 1 over 2 to the power of k.

And we have this relationship here between 2 to the power of k and the amount of money we started with, n. So the probability that I lose k times in a row and lose all my money and all my belongings is 1 divided by the amount of money I have in the first place. So perhaps, unsurprisingly, the richer I am, the less likely I am to lose all my money. And we have this really good probability given in terms of the amount of money that we start with. So if this is the probability that I lose k times in a row and lose my entire fortune, then the probability that I win (which here is winning a pound) is 1 minus this because the total probability always has to be 1.

So the probability that you will win a pound using this strategy is 1 minus 1 over n. And as n increases, this number becomes very small, so the probability gets very close to 1. (If Tom walks into the casino with a) (a hundred pounds in his back pocket, which) (I think it's conceivable -) - Okay, yeah, go with it. (And you play this strategy; how) (you're probably going to keep) (your winning of a pound?) So if I add, for n equals 100, then the probability that you win a pound will be 1 minus 1 out of 100, so it will be 99 out of 100.

So, there is a 99% chance that you will gain one pound but, and this is the key, there is a 1% chance, really small, that you will lose 100 pounds. . I don't really want to lose 100 pounds to have money, because I'm only doing it to gain a pound, is it worth it? So there's only a 1% chance, yes, but is that 1% chance of me losing my hundred pounds just to gain one pound worth it? If we do this right, I want to double the money I start with. So we need to gain a pound 100 times in a row. So we have to follow this whole strategy, with our probability of 1 minus 1 over n, but we have to do it 100 times in a row, so I'll need more paper.

The probability of winning a pound is 1 minus 1 over n. So the probability of winning 100 times in a row, which means winning 100 pounds, is just... well, I need to win the first one, then I need to win the second one, so that's just multiplying the probability and then I win the second one. third, then it's really just this to the power of one hundred. If you wanted to win 100 times in a row, it is this probability at an increasingly higher power. We're using the example of me and my hundred pounds, if I just wanted to have the probability of doubling my money, for any n, then it's just going to be 1 minus 1 over n.

Every time I win a pound I have to win n times, so it's 1 minus 1 over n to the power of n. What's going on here? So let's put some numbers. If n is one hundred, we enter this into our formula, so it will be 1 minus 1 over 100 to the power of 100, and that works out to be 0.366, so if we put it as a percentage, it will be 36.6%. So I have a little more than a third chance of getting in with my 100pounds and double my money using this Martingale doubling strategy. - (So this is a long night?) This is a very long night, yes.

Because that's... I literally have to win 100 times in a row. So I have to employ this strategy 100 times, and hopefully none of them, if I ever have a losing streak long enough to run out of money, I've already lost my £100. So I really need to go back and redo the same thing 100 times. But if I do that, I have a 36.6% chance of doubling my money using this strategy. Let's say I'm a bit more of a high roller and go for a thousand pounds, the value is 0.368. So if I have a little more money, I get a 0.2% increase.

I still start with a one dollar bet, so I can never bet more than a thousand, that's all I'm willing to put into this situation. And then if I were really big and I started and said, well, you know, I have a million pounds of capital, hopefully, then the value here is also 0.368. 36.8%. So something is happening. This is, as a mathematician, this is where you get your joy, in a way, you start thinking about these fictitious casinos, these probability problems, and then you start detecting patterns: what the hell is going on? What we're really doing here, as n increases, is like a competition between these two numbers.

Because as n gets bigger and bigger, this value in parentheses gets closer to 1 because you're taking away a smaller number. But as n grows, you're taking a number less than 1 to a higher and higher power. And when you have numbers less than 1 raised to any power, say half a square, it gets smaller. So any number less than one raised to any power, positive integer power, will get smaller and smaller. So this is always less than 1 and we raise it to a higher and higher power. So it's like a competition because you take away smaller pieces but you multiply them by itself more and more.

And that's why you get this limit, it's a real limiting process, the two balance each other out and eventually reach this constant. And the constant, and this is my favorite part of this whole problem, if you take n to infinity and do a proper limiting process at 1 minus 1 over n to the power n, this is 1 over the e-euler number, the natural rate of growth. ; the number tattooed on my arm. It's... it appears everywhere, right? Whatever you are doing with growth; It seems like anything with interest, play money, it's always e. So here is 1 over e. So the big takeaway is that if I have a fixed amount of money, n pounds, I am willing to try to double using the Martingale strategy, no matter how much money I have, the absolute best chance I have of doubling my money is about 37.%. or 1 about e.

And there's nothing I can do about it. If I get more and more money, I still have the same chance, 36.8% chance, of winning with this method. Some of you may have noticed this before or realized what's going on here. And you have to think, well, we're playing a 50/50 game, like we're literally playing red or black roulette. So we can go through all this and have, you know, the longest period in the world in a casino trying to double a million pounds and do a million iterations of the Martingale strategy, but I will never have more than 36.8% odds. to double my money.

So why don't I put everything black on the first spin? Because then I have a 50% chance of doubling my money. So, I think the moral of the story is that gaming strategies are all good and fun, but they will always fail for many reasons; One of which is that you don't have an infinite amount of money. So, although in theory we can make a pound every time, if you let the time go long enough, you will never double your money or your expected value will actually drop to zero. So, in theory, the Martingale strategy looks great.

If you do it with sensible numbers that say I want to double my money, you're literally better off putting everything in black and hoping for the best. If you want to see even more Numberphile videos with Tom Crawford, we have a playlist. There is a link on the screen and below in the video description. ..Because there is more competition to eat those rabbits. - Also, Tom will be appearing on the Numberphile podcast very soon, so stay tuned. Thanks also to Numberphile's patreon supporters. We have a full list of followers, I will include a link in the video description; but some of them also appear here on the screen; you can see their names right now.

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