Outsmarting a Coin - Or, How to Get Rich With Math

Written by Rolf

Have you always thought that tossing a coin is random? So have I. I think most people have. Some mathemagicians might disagree. I will teach you a trick that you can use at parties. Or on the streets and scam people. If you choose to use this trick to get rich, I want part of your profit.

The game is about tossing a coin several times and guessing a sequence. Sounds like a fair game, right? Wrong. I will explain the exact bet, after which you will probably still be convinced it's fair. More importantly, your opponent will think it is fair. I will include a script I wrote to prove it is not exactly fair, but first I will explain the bet.

Note: this assumes a perfectly fair coin with equal chance of either head or tail. It's not a magic trick; it is math.

Your friend (opponent, victim, whichever your intentions) chooses a sequence of 3 coin toss results, for example "head tail head". Then you will also choose a seemingly random sequence of 3 tosses. Then you (or your friend) start tossing a coin. The one whose sequence occurs first, wins. Still sounds fair, doesn't it? Good, because it's not.

To prove it is not fair, I have written a little script to demonstrate this. Because you still think it is a fair bet, you will be the victim. Enter a 3-toss sequence in the field that has a red « pointing at it. Type something like "HHT" or "THH" or "THT" or whatever you want. After you have chosen a sequence, press the button labeled 'Go' and the trickster (the CPU) will choose its toss series. Then a third, unbiased virtual person will start tossing a fair coin (a simple random number, 0 or 1, that represents head or tail). The game ends when either your or the trickster's sequence is tossed, and the score is updated. Do this several times, to get a good statistical impression of the results. You will usually start to realize the unfairness after 20 or 30 tosses. Pressing 'New game' will reset the scores and start a new simulation.

Have a go at it, and then I will explain how to stop being a victim and perform the trick yourself.

	You:	CPU:
Your sequence:	«
CPU sequence:
Coin sequence:

Part of the trick is that you let your friend choose the first sequence. If you want to look completely ridiculous, scratch your beard, utter some random floating point numbers and mumble something about solar flare and the earth magnetic field before announcing you have come up with a better sequence. Note: this act requires a beard.

I will teach you how to determine a better sequence than your opponent. His first and second guesses will be your second and third. Your first guess will be the opposite of his second guess. For example, if your friend chooses HHH, you will choose THH. If your friend chooses THT, you will choose TTH. There are only a limited number of possibilities, so I will just list them all here, together with the odds of you winning:

victim	you	odds
HHH	THH	7:1
HHT	THH	3:1
HTT	HHT	2:1
HTH	HHT	2:1
THH	TTH	2:1
THT	TTH	2:1
TTH	HTT	3:1
TTT	HTT	7:1

This 'phenomenon' was discovered in '69 by mathematician Walter Penney and is known as a non-transitive game. Penney deserves a few percent of your profit, too. If you can't find him, send it to me and maybe I will make sure he gets his share.

Another typical non-transitive game is rock-paper-scissors, but there it is immediately obvious that you can find a winning answer to the victim's proposition, were he to announce it. This coin-toss game is way more subtle, and seems to go against your instinct and everything you've learned in school.

The sequence of tosses can be longer than three, but usually it's 3 because it's simply easier. It is vital that the victim makes up and announces his sequence first. Based on his guess, you can determine a 'better' sequence; a sequence that has a bigger chance of occurring first. This sounds strange (especially when you've drunk a glass of vodka, as I did last night) but consider this: if the victim chooses TTT, you will choose HTT. If any of the (first three) tosses yields head (7 in 8 chance, or 87.5%), the victim can not win anymore.

Except Mr. Penney, credits also go to my good friend Dramed, who told me about this magic trick that isn't a magic trick. At first I thought he was totally delirious and I refused to even test it out, until I had read some more about the math behind it. So, here's to you, Dramed ;)