MAIN CONCEPTS
From this problem, we can experimentally get value of pi from
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This puzzle certainly deals with the aspect of probability, something many of you should be familiar with. Let’s have a small discussion about it. Say you have to close your eyes and point to one of the sections of the wheel below.

Your chances of pointing to a black one is . Why? Simply because there are 4 different sections, or 4 different outcomes and out of those 4, only 1 of them is black. Now, what are your chances on pointing to a black one with the wheel below.

It is now . 3 black sections out of 8 available outcomes. Simply enough to understand.

Now, how about this needle problem? Obviously, it seems hard to visualize what is the total possible amount of outcomes. To help us, we create a coordinate system. Define  and  as shown below where  is the distance OP from the midpoint of the needle to the nearest crack, and  is the smallest angle between OP and the needle.
 
Notice that a random toss of the needle can be ‘marked’ out by this new coordinate system with the variables in the interval

 and

which covers all the random positions the needle can fall to. Further, notice that the outcome with are interest in can be written as the inequality

By plotting the graph  as follows,

 

we see that this inequality describes the shaded region under the graph. hence, we conclude that the probability of the needle falling across a crack equals to the following ratio of areas:

which is slightly less than .

Who would have thought that a question starting out on probability could have used the integral calculus to solve it.

Here is an experimental procedures to think about. Say that I decided to carry out the experiment of tossing and recording the amount of times it falls across the crack. I toss the needle  times and record when it falls  times. I happen to be very free to the point where I performed this experiment an infinite amount of times. Mathematically then, we have

which reduces to

assuming I carry out the experiment infinite amount of times. And by doing so, I have just found the value of , or at least a close approximation by theory, without any use of geometry let alone circles. Now you can tell your friends how to get its value without wrapping a string about a tennis ball, which was how I did it in middle school.