a person familiar with the laws of mechanics can toss a biased coin so that it will produce predominantly either heads or tails, at will. [...] From the fact that we have seen a strong preponderance of heads, we cannot conclude legitimately that the coin is biased; it may be biased, or it may have been tossed in a way that systematically favors heads. Likewise, from the fact that we have seen equal numbers of heads and tails, we cannot conclude legitimately that the coin is ‘honest’. It may be honest, or it may have been tossed in a way that nullifies the effect of its bias.
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An important feature of this tumbling motion is conservation of angular momentum; during its flight the angular momentum of the coin maintains a fixed direction in space (but the angular velocity does not; and so the tumbling may appear chaotic to the eye). Let us denote this fixed direction by the unit vector n; it can be any direction you choose, and it is determined by the particular kind of twist you give the coin at the instant of launching. Whether the coin is biased or not, it will show the same face throughout the motion if viewed from this direction (unless, of course, n is exactly perpendicular to the axis of the coin, in which case it shows no face at all). Therefore, in order to know which face will be uppermost in your hand, you have only to carry out the following procedure. Denote by k a unit vector passing through the coin along its axis, with its point on the ‘heads’ side. Now toss the coin with a twist so that k and n make an acute angle, then catch it with your palm held flat, in a plane normal to n. On successive tosses, you can let the direction of n, the magnitude of the angular momentum, and the angle between n and k, vary widely; the tumbling motion will then appear entirely different to the eye on different tosses, and it would require almost superhuman powers of observation to discover your strategy. Thus, anyone familiar with the law of conservation of angular momentum can, after some practice, cheat at the usual coin-toss game and call his shots with 100% accuracy. You can obtain any frequency of heads you want – and the bias of the coin has no influence at all on the results!
Unless the person throwing read Jaynes:
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