A Brief Journey in Discrete Mathematics by Randolph Nelson

Author:Randolph Nelson
Language: eng
Format: epub
ISBN: 9783030378615
Publisher: Springer International Publishing

6.2 The Correct Insight

At the beginning, the mathematical properties of coin tossing were obvious and intuitive. Analysis then abruptly transformed our facile conclusions into a quagmire of perplexity. We are at the point of puzzlement to ask the simple question: what is really going on? The answer is straightforward. First be assured that the above analysis is correct. The game does behave in the somewhat neurotic way outlined by the previous equations. The game is not at fault, coins do not conspire to fool us. Shakespeare says it best; in Julius Caesar, Cassius opines,

The fault, dear Brutus, is not in our stars, But in ourselves, that we are underlings.

We need to look no further than ourselves to find the glaring fault. Common sense notions of coin tossing are just dead wrong. The root of the misunderstanding arises from the fairness of the game. Because there is no bias for either player to win or lose a toss, there is no preferred direction for the game. This directionless aspect of coin tossing subtly violates intuition.

If there was the slightest bias in the coin to make the outcome of the game more even, then all of the intuitive comments made throughout the chapter would hold. For instance, suppose the coin became slightly biased towards landing up tails when player 1 was ahead. Also assume a similar bias towards landing up heads if player 2 was ahead. Under these assumptions, even if this bias was slight, then the intuition found in the opening remarks of the chapter would be satisfied. Without such a bias, there is no tendency to even out the players fortunes and the game simply drifts aimlessly along without any tendency to make things come out fair and even. At each toss, the game effectively starts anew from its current level. One player will be more lucky than the other. It is that simple; intuition lies on a knife edge, violated if completely balanced. It is ironic, that it is exactly the notion of fairness that produces outcomes that seem unfair. Having one player be ahead of the other most of the time, as we have just seen, is the game’s natural state.

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