Am I Being Random, Still?

What if we tossed a coin ten times and got

HHHHHHHHHH

and then we asked you to bet on the next toss?

Ten Hs in a row has a probability ½^10= 0.0009765625. Pretty unlikely. But any mathematician will tell you that this sequence, or a sequence of alternating H and T, or Hs and Ts representing the digits in the square root of two, or in pi, or any sequence whatsoever, are all equally likely (assuming an unbiased coin). Or rather, equally unlikely. But it’s hard to get one’s head around this fact. One can’t help feeling that if you were asked to bet after ten Hs then you should be suspicious.

Some people think that the Law of Averages applied here means that after so many Heads the next toss is more likely to be Tails to “balance things out.” This is called “The Gambler’s Fallacy.” The Law of Averages is a layman’s version of the more mathematical Law of Large Numbers. And is commonly misunderstood. In a nutshell, the Law of Large Numbers says that after a large number of trials, here tosses, the average should be close to the expected value, and get closer as the number of trials increases. If Heads counts as plus one and Tails as minus one then the expected value is zero for an unbiased coin. As the number of tosses increases so the average, the sum of the plus and minus ones divided by the number of tosses will converge to the expected value of zero. But this says absolutely nothing at all about the next toss, which will always be equally likely to be Head or Tail.

Casinos know all there is to be known about probability. And the Law of Large Numbers is tattooed on their black hearts in red ink. They know for example that each spin of the roulette wheel is independent of all previous spins. They know that ten Reds in a row at roulette is as likely to be followed by a Black as another Red. But they also know that many people don’t believe this simple fact of probability, and physics. They know about apophenia, that people see non-existent patterns, and that they play systems. And these are the people casinos adore, people who believe they can beat the casino and bet accordingly. That is why they will often encourage such people by presenting a list of recent numbers, printed out or on electronic signs near the wheel. Such data is also available online. A quick search will show you people discussing roulette patterns in all seriousness. Such people are the suckers that casinos rely on for business.

You can’t win at roulette.

Or can you? More anon.

In our ten-Heads-in-a-row example maybe the sequence is genuinely random, and it’s 50-50 what the next toss will be. Or maybe the coin is biased, or double headed. Or maybe you’re being lured into thinking it’s double headed and the next toss will be a Tail. On balance, this is a bet best avoided.

We’ve seen something not unlike this in the world of finance only a few years ago. The returns of Bernie Madoff’s fund. The S&P500 index goes up and down, then down and up, down down up up down, and so on, in what looks, and probably mostly is, a random fashion. (Even if the technical analysts think they can see patterns.) On the other hand Madoff’s returns go up, and up, and up, and… up,… in the hedge fund equivalent of a never-ending sequence of Heads. Too good to be true? Yes, and you’d be right to be suspicious.

Derren Brown gives the following wonderful performance. Picture this…

Derren stands on the stage with his goatee beard, suit from days of yore, and a microphone. He asks all members of the audience to stand and put a coin or other object into one of their hands. He says “Left” and everyone with the object in their right hand is asked to sit down. He repeats this several more times. Each time approximately half of the audience is asked to sit. He is down to three audience members still standing. Clearly these are all people with whom he has a “connection.” One of these he chooses and asks to join him on stage.

Derren asks her — in the youtube video of this the audience member is female — to put her hands behind her back and put a coin in one hand. She then holds her hands out in front. Derren has to say which hand the coin is in. His typical patter goes like “Last time you put the coin in your right hand so this time you think I’ll think you’ll put it in your left hand. But you know I’m thinking this so you’ll put it in your right again. But you know I know what you’re thinking so you’ll put it in your left. Forget that, actually your right hand is, you’ll notice, slightly lower than your left. And that’s because you are over compensating for the weight of the coin. Left!” And he’s right!

He does this many, many times in a row, each time correctly figuring out which hand the coin is in.

How does he do it?

Maybe it’s luck. Unlikely. One half to the power of… Or the audience member could have been a plant. Too easy, and also reputation harming. Maybe by pruning the audience he’s found someone who thinks like him. Maybe there’s some psychology going on. We’ve seen online discussions of all this. It’s all about Neuro Linguistic Programming was one suggestion. For example, looking up and to the left is, according to NLP, “Non-dominant hemisphere visualization i.e., remembered imagery.” So DB is looking at the person’s eyes for clues perhaps. Unfortunately NLP seems to have been largely discredited. Another suggestion was that some people were just easy to read. This was from the poker players, who told of easy-to-read poker “tells.”

Or maybe it’s a trick.

How it’s really done we won’t tell. Let’s just say it will set you back one of either a) many years of dedicated practice or b) a couple of hundred dollars. But the most important thing we gleaned from said discussion was how keen people were to believe in (pseudo) science rather than, ahem, perhaps trickery. Homeopathy, crystals, crop circles…just mention energy and vibrations and how some German scientist has proved everything while living on a diet of carrot juice and there’s a decent chunk of society that will believe you. Even smart people.

We tried to get DB to perform at one of our book launches. But DB’s fee of £30k for 40 minutes was, er, a bit steep for our publisher. And this was just as DB was becoming famous. Lord knows what he charges now.