It's not
that curious, throw in a little
calculus and
everything makes sense.
The root of the problem is that we need to find the
probability that a
needle will cross a line. For simplicity, we will assign the
length of the needle and the distance between the lines a value of 1. As the needle is dropped, it can land at any
angle. Assuming the lines run vertically, the maximum
horizontal span of the needle, 1, occurs when the angle between the needle and the vertical is π/2
radians. The minimum span, essentially 0, occurs when the needle
falls parallel to the
lines, so that the angle between it and the vertical is 0.
(If you're not into
geometry, π/2
radians is the same as 90
degrees.)
Applying some
trigonometry, we can easily show that the horizontal span of the needle is given by sin(x) where x is the angle the needle makes with the vertical. The next step is to find the average span, that is, the
average value of the function
sin(x) as x varies between 0 and π/2
degrees. From
calculus, this is defined as:
π/2 /
∫ sinx dx /
0 / π/2 - 0
/
(Or if you don't like
calculus, it is the area under the curve
sin(x) divided by the length of the base of the
curve, in this case π/2.)
Evaluating this
integral, we get
|π/2
-cos(x)| = cos(0) - cos(π/2) = 1
|0
Dividing by π/2 - 0, we get 2/π. This is the average
horizontal span of a
needle dropped on the
vertical lines. Without going into a proof, we can show by common sense that the probability that a needle will touch a line equals its
average span divided by the distance between lines (which we defined as 1, so the
probability is just 2/π). (Just think, as span approaches the distance between lines, this ratio approaches one, or maximum
probability, so as span approaches the
average span, the
ratio goes to average probability.)
By definition, probability is the number of needles that cross a line (c) divided by the total number of needles (t) Therefore:
P = c/t = 2/π
Which can be rearranged to π = 2t/c
As with all probability problems, the more trials involved, the better the estimate becomes.
Note, I didnt mean to ruin the mystery of it...in my
opinion the
beautiful simplicity of this makes the
world an even
more interesting place.