复旦大学：《概率论》精品课程教学资源（习题答案）布丰投针问题 Buffon’s needle problem

Gaussian Technologies Department of Mathematics Calculus related puzzles Buffon's needle problem A needle 2 inches long is dropped at random onto a floor made with wooden boards 2 inches in width placed side by side. What is the probability that the needle falls across one of the cracks? 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 enoug to understand MISCELLANEOUS CALCULUS RELATED PUZZLES

Gaussian Technologies Department of Mathematics MISCELLANEOUS > CALCULUS RELATED PUZZLES Calculus related puzzles Buffon’s needle problem A needle 2 inches long is dropped at random onto a floor made with wooden boards 2 inches in width, placed side by side. What is the probability that the needle falls across one of the cracks? 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 1 4 . 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 8 . 3 black sections out of 8 available outcomes. Simply enough to understand

Gaussian Technologies Department of Mathematics 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 x and 6 as shown below where x is the distance OP from the midpoint of the needle to the nearest crack, and 0 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 0≤x≤1and0≤≤ 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 x<cos e By plotting the graph x= cos 0 as follows MISCELLANEOUS CALCULUS RELATED PUZZLES

Gaussian Technologies Department of Mathematics MISCELLANEOUS > CALCULUS RELATED PUZZLES 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 x and ! as shown below where x 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 0 ! x ! 1 and 0 ! " ! # 2 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 x < cos! By plotting the graph x = cos! as follows, P x cos! ! 1 O

Gaussian Technologies Department of Mathematics x=cose 6 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 cos 8de area under curve area of rectange 兀/2 兀/22 which is slightly less than 3 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 n times and record when it falls k times. I happen to be very free to the point where I performed this experiment an infinite amount of times. Mathematically then we have k 2 n→n兀 which reduces to MISCELLANEOUS CALCULUS RELATED PUZZLES

Gaussian Technologies Department of Mathematics MISCELLANEOUS > CALCULUS RELATED PUZZLES 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: area under curve area of rectange = cos! 0 " /2 # d! " 2 = 1 " 2 = " 2 which is slightly less than 2 3 . 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 n times and record when it falls k times. I happen to be very free to the point where I performed this experiment an infinite amount of times. Mathematically then, we have limn!" k n = 2 # which reduces to ! 2 x ! 1 x = cos!

Gaussian Technologies Department of Mathematics 分小、 k assuming i carry out the experiment infinite amount of times. and by doing so, i have just found the value of t, 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 MISCELLANEOUS CALCULUS RELATED PUZZLES

Gaussian Technologies Department of Mathematics MISCELLANEOUS > CALCULUS RELATED PUZZLES k n ! 2 " 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