I’m going to talk to you about the type of maths I like. The type of maths I like is probability theory. My story is about my friend Daan. He likes to brag that he always wins money at the casino or playing with free spins no deposit win real money in online casinos. Allow me to explain Daan’s strategy.
Daan says that he plays roulette in the casino. With roulette, you have a table with a sectioned layout. A ball is spun, falls into a pocket… …and this pocket is either red or black, or even green.
You can bet on all kinds of things. If you bet red, you win when the ball ends up in a red pocket. If you bet black, you win when the ball ends up in a black pocket. You bet a certain amount on either red or black…and if you win, you receive double that amount.
If your stake is 1 euro, you win 2 euros, and so your profit is 1 euro. If you lose, your stake is gone. Daan’s strategy goes as follows: I go there, I keep doubling my stakes, until I win and then I stop. Let’s see what happens, because he says this is making him rich. He’s not rich, so that’s suspect in itself. Let’s see.
In the first round, his stake is 1 euro. This means that when he wins, he has a profit of 1 euro on this side. But if he loses, he’s on -1 euro. In the second round, he doubles his stake, which amounts to 2.
If he loses, 2 is subtracted from -1, and so he goes to -3. If he wins, 2 is added to -1, and so his profit is 1 euro. In the third round, his stake is 4 euros, double the previous stake of 2 euros. If he wins, 4 is added to -3, which results in 1. If he loses, he’s on -7. One more round to clearly show the pattern here.
His stake would be 8 euros. If he wins, 8 is added -7, resulting in 1. If he loses, 8 is subtracted from -7, which puts him on -15. Oh dear, there’s ink everywhere. This means… You see: as soon as Daan wins, he makes 1 euro profit.
He could just continue doing that until he’s rich. Daan’s argument is that he has to win sooner or later. He can’t just keep losing several rounds in a row. For example, one can easily play 100 rounds of roulette in a couple of hours. What are the chances of losing 100 rounds in a row? We can calculate this.
For simplicity’s sake, we’ll assume that the chance of red winning is exactly 0.5. It’s actually a bit lower. And for black it’s also 0.5, making it easier to calculate. This means that losing 100 times in a row… The chance of losing is 0.5.
Multiply that by 0.5 for losing the next round. 0.5 times 0.5 for losing twice in a row. The chance of losing three times in a row is 0.5 times 0.5 times 0.5. Losing 100 times in a row, is expressed by 0.5 to the power of 100… …which is equal to 2 to the power of -100. Or put differently: Approximately 7.8 times 10 to the power of -31.
That’s a really low number. I’ll give you an idea of how low exactly: The time the universe has existed, in seconds… …is approximately 10 to the power of 17. If you had a played a round of roulette for every second since the Big Bang… …there’s a very low probability you would’ve lost 100 times in a row. This is what Daan’s strategy is based on. Sooner or later he has to win, because losing 100 times in a row is impossible.
But as said, Daan still isn’t rich, and so there’s something wrong. Let’s calculate what Daan’s stake is in round 100. Stake…
I’m sorry, that should be total stakes. We know that the stake in round 100 is 2 to the power of 99. But we want to know the total of cumulative stakes after 100 rounds. In the first round the stake was 1 euro. We assume he plays 100 rounds.
In the second round it was 2 euros, in the third it was 4 euros… …and so on, all the way to 2 to the power of 99. Here we have 2 to the power of zero, to the power of 1, to the power of 2… …and so at round 100, it’s 2 to the power of 99. How do I add all of this? I’m applying a trick, as mathematicians are wont to do.
I’m going to multiply by 1. That sounds as if nothing will change, but you have to do it the smart way. In this case, you multiply with 2 minus 1. So, it’s 2 minus 1, times 1 plus 2 plus 4, and so on… …all the way to 2 to the power of 99. What does it say?
2 times 1 is 2, 2 times 2 is 4, 2 times 4 is 8… …and 2 times 2 to the power of 99, 2 to the power of 100. Now for the -1. That’s easy.
It’s exactly the same as what we had. The big advantage of doing it this way, is that I can now cancel out numbers. 2 against 2, 4 against 4, 8 against 8… …2 to the power of 99 against 2 to the power of 99. What remains?
This number and this one. That’s 2 to the power of 100, minus 1. 2 to the power of 10 is approximately 1000. 2 to the power of 20 is around a million.
2 to the power of 30 is a billion. 2 to the power of 100 is more money than there is on the whole planet. Daan will never be able to bet such a stake.
Casinos are smart enough to say that you need to have the money you want to bet. That means that at some point the jig is up for Daan. He’s not making it to round 100… …or even to round 20, as he’s not a billionaire. Let’s say he can play M rounds, with M representing a number. That leads us to the following calculation: In round M… …his stake is 2 to the power of M minus 1.
He can do two things, either win or lose. If he wins, he gets that 1 euro profit, after all. But he can also lose.
If we perform the same calculation as this one here… …we see that his total becomes -2 to the power of M… …minus 1, or rather plus 1, because this here is already minus. What is his expected profit after M rounds? He can either lose M times in a row, and in that case…
The probability that he loses M times in a row, is 2 to the power of -M. How much will he have lost then? His profit will be negative… …and amounts to -2 to the power of M, minus 1. But he can also have won, somewhere between round 1 and round M. In that case, he has won 1 euro. What’s the probability?
It has to be 1 minus the probability that he loses M times. That’s 1 minus 2 to the power of M. Let’s write that out. It’s -1 plus 2 to the power of -M… …plus 1 minus 2 to the power of -M, and that equals zero.
So, the expected profit after M rounds, is simply zero. Which explains why Daan isn’t all that rich. You might say that Daan isn’t smart and needs to devise a better strategy. Unfortunately, that can’t be done.
In a masters class I teach, we prove the following statement: If you play a betting game… …where your expected profit without a strategy is zero… …the expected profit after M rounds with a strategy will also be zero. Likewise, if the expected profit is negative, which it is in a casino… …the expected profit after M rounds with a strategy will also be negative. Does this mean it doesn’t matter whether you use a strategy in the casino or not? It depends on how you look at it.
You can do interesting stuff with a strategy. For instance, you can play with 32 people on the same roulette table. 32 equals 2 to the power of 5.
That allows you to cover all possible outcomes of 5 rounds of roulette… …ignoring the green pocket. That means person 1 bets on red five times in a row. Person 2 bets on red four times in a row, and then black.
Person 3 bets on red three times in a row, then black and then red. Person 4 bets on red three times in a row, then black and black again. You do this with all 32 persons, to cover all possibilities. This means that after those 5 rounds… …one of those 32 people will have won 5 times, and one will have lost 5 times. If everyone hadn’t agreed to do this… …the probability that one of those 32 people wins 5 times in a row… …would’ve been smaller than 1 divided by E, which is smaller than half… …or smaller than 1, either way. So, for a group with a strategy, the large ups and downs… …the largest profits and losses within that group… …are potentially larger than those of a group with no strategy.
It’s not a coincidence that I’m telling you about this. This is related to the research I’m doing. In my research, I look at models for systems… …that are characterized by arbitrariness in many spots. For example, an emulsion of oil droplets in water. Those droplets aren’t perfectly round. The edges are somewhat arbitrary, going every which way.
Shares are another example. If you look at share prices, you don’t see straight lines… …but rather, they seem to be going up and down arbitrarily. Often, the average size of such a wobble… …of such an arbitrary step, depends on the state of the system. If I look at a share price and it’s very high… …the average jumps will be larger… …than when the price is low. One interpretation is that the share has a gambling strategy. What happens, depends on the state you’re in.
It’s quite difficult to analyse. What we’d like to understand… …is whether one can compare such a system with a system with no strategy… …where the rate of change doesn’t depend on the state you’re in. This kind of questions are related to that. If you enjoyed this and want to know more about maths… …check out our study programme at the University of Amsterdam… …via the link below. Maybe we’ll see each other in class.
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