By Krauth W.
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5 The Ultimate Algorithm Cutting up the field into little squares allows us a second time to make the program run faster by a factor N , where N is the number of little squares. Not only can we use the squares to simplify the calculation of overlaps, but to exclude large portions of the field from the search. 38 Unfortunately, you will quickly find out that the program still has a very large rejection probability . . 3: roughly 2% of the square’s surface can only accept a new coin. So, you will attempt many depositions in vain before being able to do something reasonable.
In fact, you can predict analytically what will be the distribution of time intervals between “flips”. For the little boy, at any given time, there is a probability of 5/6 that one roll of the die will result in a rejection, and a probability of (5/6)2 that two rolls result in successive rejections, etc. 3. You can easily convince yourself that the shaded space in the figure corresponds to the probability (5/6)2 −(5/6)3 to have a flip at exactly the third roll. So, to see how many times you have to wait until obtaining a “flip”, you simply draw a random number ran 0 < ran < 1, and check into which box it falls.
B 46 11190 (1992)  M. A. Novotny Computers in Physics 9 46 (1995)  M. A. Novotny Phys. Rev. Lett.