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I was walking past a tree when I thought about a problem which I've been trying to solve.

It states that "If there are 20 leaves on a tree and all the leaves fall on the floor, find the probability that one leave will fall pointing directly north" . Does anyone have any ideas of how to attack this problem?

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    One has to create a model. It is reasonable to assume that the orientation of the leaf has a continuous distribution. Then the answer is $0$. If you are willing to accept in margin of error in the interpretation of "directly north" one can find an answer. I assume that "one leaf" means at least one. For simplicity assume angle the tip makes with true north is uniformly distributed. Then if you decide that within $5$ degrees is close enough, you should be able to find an answer.2011-05-24
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    "I know it's similar to the buffon needle problem (somewhat) but it seems more complicated." On the contrary, it's unrelated to the Buffon needle problem, and much simpler. The probability is 0.2011-05-24
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    OK. I now see what you mean.2011-05-24
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    This is analogous to asking, "If I pick a real number between $0$ and $1$, what is the probability that I pick exactly $1/2$?"In general, if you have a continuous probability distribution over $\mathbb{R}$, the probability of any particular number is $0$ (although is is possible to have a probability measure with atoms, but then the cumulative distribution function isn't continuous).2011-05-24

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Unless you allow some tolerance around directly north, the chance is zero. If you allow a tolerance (plus or minus) of $\epsilon$ the chance for one leaf is $\frac{\epsilon}{\pi}$ and the chance of at least one of twenty is $1-\left(\frac{\pi-\epsilon}{\pi}\right)^{20}$. It's just like flipping a coin.

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    +1. Replace $\pi$ by 180 if you think in degrees rather than radians.2011-05-24