Ok, this is an eighteenth century problem first proposed by Georges-Louis Leclerc, Comte de Buffon (Yes that is only one name!). Suppose a 2 inch needle is randomly thrown or dropped onto a floor made up of wooden boards which are also 2 inches wide and are placed side by side. Now the question is what is the probability that the needle falls across one of the cracks?

Ok, let us try to derive a probablistic formula for this. We need to consider the total possible outcomes of the random experiment. First we need to create a coordinate system:

Where ‘x’ is the distance OP from the midpoint of the needle to the nearest crack and ‘theta’ is the smallest angle between OP and the needle. Note that half the needle has length 1 inch. Notice that we can mark out a random toss of the needle by constructing a coordinate system with the appropriate variables satisfying certain conditions, i.e. the angle ‘theta’ must be between zero and (pi/2) inclusively and the other variable ‘x’ is between zero and one inclusively.(notice we are using radians here) These conditions cover all possible positions that the needle can fall to. We can further notice that the outcome of interest can be thought of as the variable ‘x’ being strictly less than the cosine of the angle ‘theta’. Now plotting the graph of x = cos (theta) with the intervals just deifned above we have:

And we can see that our outcome of interest describes the region under the graph (above). So now we conclude by considering the ratio: area under graph/area of rectangle. We calculate the area under the graph by integrating our function (cosine) with limits from zero to (pi/2) with respect to ‘theta’. This gives us an answer of 1. And the area of the rectangle is simply (pi/2). Now considering the ratio we have:

area under graph/area of rectangle = 1/(Pi/2) = (2/Pi). This gives us the probability of the needle crossing a crack. Say that I carry out this experiment of tossing the needle and recording how many times it crosses a crack. More precisely say that I toss the needle ‘n’ times and record it ‘k’ times when it crosses a crack. Furthermore consider I do this experiment an infinite amount of times, then mathematically we have:

.  Then assuming that I carry out this experiment an infinite number of times, we can reduce this to:   . And so you see we have found the value of ‘pi’ or a close approximation to it, just using a needle and wooden boards and without no use of geometry i.e. using circles. Note this problem can be done with matchsticks and lined paper as long as it satisfies the conditions of the experiment. This is an illustration of the beauty and mystery of mathematics. By this I mean we used calculus to solve parts of the problem, and then ideas of limits. Beautiful! This problem can also be solved by integral geometry through methods of multivariable calculus.