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Coin Flips Are Biased

In a shocking (shocking!) development, Persi Diaconis, Susan Holmes, and Richard Mongomery have published a convincing theoretical model that shows that, even with idealized physics, the geometry of a coin flip inherently favors the side that starts on top.

The professors explain this bias by drawing our attention to the axis of rotation of a tossed coin.

The Geometry of a Coin Flip

Imagine a flipped coin that starts perfectly flat heads up, then rotates on an axis that is exactly coplanar with the coin. During the coin flip the normal vector in the direction of heads will trace out a perfect circle that crosses through both the "north pole" (where it starts, straight up) and the "south pole" (straight down). The traced great circle is half in the top hemisphere and half in the bottom, and so with this rotation geometry, the likelihood of heads is an even 0.5.

But a flipped coin is not constrained to rotate on a perfectly coplanar axis, and that is the key. When there is a non-coplanar axis of rotation, the "heads" normal vector traces out a smaller circle, centered on the axis of rotation. The traced circle will still pass through the north pole (because that's where it starts) but it will now miss the south pole. The center of the circle will be somewhere above the equator.

The Bias Depends on the Axis of Rotation

Since the traced circle will have more of its circumference in the upper hemisphere than the lower hemisphere, the probability of the normal ending up in the upper half - and thus the probability of heads - exceeds 0.5. The amount of bias depends on the exact orientation of the axis of rotation.

If the axis of rotation is more than 45 degrees away from the plane of the coin, the coin will never turn tails up, no matter how vigorously it appears to flip when observed in the air. Intentionally tossing a coin with this geometry is a good magician's technique.

The professors derive exact an formula for the probability function as a function of rotation angle in their paper.

Experimental Verification

The professors also execute two clever experiments where the deviation of the axis of rotation from the plane of the circle is measured in the real world, to see if it actually deviates from the plane during an ordinary coin flip. While one of the experiments relies on high-speed photography and computerized image processing, the other experiment involves scotch tape, a thin ribbon, and a half dollar.

That second experiment is begging to be reproduced and verified at other research institutions around the world or in your living room.

Unfinished Science for Future Generations

Although the professors report that their experimental results are consistent with their theoretical model, the bias was not directly observed. The professors estimate that a direct verification of the predicted bias would require 250,000 real-world coin flips, a task that they did not attempt.

And so the path to scientific enlightenment never ends. The herculean work of a direct observation of the predicted coin-flipping bias is left for future intrepid researchers.

Posted by David at August 27, 2009 11:12 PM