"Most" of the needles that cross a horizontal line are oriented in a "nearly vertical" direction. In other words, once we know that a needle is one that crosses a horizontal line, that gives us a clue that its orientation is more likely to be fairly vertical and therefore that it is less likely than usual to be a needle that crosses one of the vertical lines. Vaguely related to the Will Rogers "paradox", Monty Hall problem, or probably about a dozen others. On Thu, Nov 25, 2010 at 08:26, Veit Elser <ve10@cornell.edu> wrote:

When Buffon's needle is dropped at random on parallel lines spaced by the needle's length, it crosses a line with probability 2/pi.

Add another set of parallel lines, perpendicular to the first, and with the same spacing. The probability that the needle crosses two lines, one from each set, is not equal to (2/pi)^2, as it would if the two types of crossing were independent, but 1/pi.

So there is the curious fact that the probability of crossing a horizontal line and a vertical line is equal to the probability of crossing a horizontal line and not crossing a vertical line (since their sum must be 2/pi). Equivalently, the conditional probabilities, of crossing or not crossing a vertical line, given that a horizontal line is crossed, are equal.

If you believe there are no accidents in mathematics, then surely there is some bijection that explains this fact ...

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