1) The fair bet or a completely fair game.
This is also an invention of mathematics. Finding this game in practice is equally impossible. Even the toss of a coin for equal money would not be an absolutely fair game. The coin cannot be balanced at all, one side is heavier or lighter or the centre of gravity is not in the middle (where would that be anyway?). And then there are the usual questions: who throws, on what surface and who has what intention, etc.? So it’s more of a theoretical game. But here it’s interesting that, at least theoretically, there is always the fair or correct payoff, so to speak. When rolling the dice, if you bet on a number, it would be 6:1 or 6.0. 100 euros bet, 600 euros payout if successful, that means 500 euros profit. So 5 times you lose 100 euros, once you win 500 euros. That’s what I call fair. In the long run, it should at least be in the order of magnitude: 5 times your number doesn’t come up, 1 time it does. The bottom line is 500 DM lost and 500 DM gained. And if someone has an advantage, those involved (usually) don’t know it.
But I’ll give you another really interesting example of how you can make an exciting game with a slight modification of the game, in which there may already be a share of playing strength: One of the two players takes the coin in one hand. The other must try to guess which hand it is in. Then the sides are switched (or not, it wouldn’t have to be). The consequence: Would it then be a game of skill? Does one side possibly already have an advantage through the ability to read the other’s mind better? “He’ll probably want to trick me now, and he’ll take the coin back in the same hand, then again.”
So, as far as I know, this game is not played, except as shing shang shong, where you have three options and you only have to figure out unimportant decisions. But it would be interesting. But I have the feeling that people shy away from it (I played the game once with a classmate when I was at school. Without any stakes, of course. But actually we also wanted to prove to ourselves how well we knew each other. And in fact we both got far more than 50% hits. Coincidence?).
But I also have a solution to offer: Should you get involved in this game, I can promise you at least 50% with a simple trick: Secretly toss the coin before each attempt, into which hand you should take it. Then all the psychological tricks will be of no use to your opponent. He cannot achieve more than 50% in the long run. However, you are also guaranteed not to achieve more than 50%. So if you want to play this game with an advantage, you have to try to delve deep into your opponent’s psyche.
In this context, another problem comes to mind that I have encountered many times. It sounds so simple, but I haven’t really found the right solution yet. Maybe you can come up with something guaranteed to work? The question is as follows: We play a game of chess. No problem, we’ll draw lots for the colours. We’ll draw lots for the colours. But we play the game by telephone. Who has white? Who gets to start? How do we draw lots? Fairly and correctly?
The only solution I ever came up with was like this: First, we agree who will take the even numbers and who the odd numbers. The winner, of course, has the choice of colour. Both people say a number at the same time, preferably between 0 and 9, the 0 has to be there, because there has to be an even number to choose from. These two numbers are added up. The sum is even or odd. 0 itself is even. One of the two is the winner. Problem only, somewhat related to the shing shang shong problem, where one occasionally sees the other’s hand open (or close, to the stone), and makes or varies one’s decision based on that: When and how is simultaneous? And: hearing and speaking at the same time can also lead to misinterpretation. “What did you say? But I heard two.”