Particularly in many disparate natural datasets and mathematical sequences, the leading digit ( d) is not uniformly distributed but instead has a biased probability P( d) = log 10(1 + 1/ d) with d = 1, 2, …, 9, known as Benford’s law. The distribution of prime numbers is essential to mathematics as well as physics and biology. Lackmann ( Springer-Verlag, Berlin, Heidelberg, 2011). Tao, in An Invitation to Mathematics, edited by D. The study of the distribution of prime numbers has fascinated mathematicians and physicists for many centuries. If the last digits of prime numbers come out with the same frequency, then the probability of the four last digits would be equal, i.e., prob( j) = 25%. In mathematics, the last digits are believed (without a proof) to be random or evenly distributed when numbers are large enough. All primes except 2 and 5 should end in a last digit ( j) of 1, 3, 7, or 9. Prime numbers are positive integers larger than 1 they are divisible only by 1 and themselves. Ī similar situation appears in distribution of last digits in prime numbers. coin or dice tossing is commonly believed to be random but can be chaotic in the real world. The randomness in coin tossing or rolling a dice is of great interest in physics and statistics: 8–13 8. Making a choice by flipping a coin is still important in quantum mechanical statistics. In fact, real coins spin in three dimensions and have finite thickness, so coin tossing is a physical phenomenon governed by Newtonian mechanics. This situation is valid only under a condition that all possible orientations of the coin are equally likely. For a fair coin, the probability of heads and tails is equal, i.e., prob(heads) = prob(tails) = 50%. It is commonly assumed that coin tossing is random. ![]() ![]() Coin tossing is a simple and fair way of deciding between two arbitrary options. by flipping a coin, one believes to randomly choose between heads and tails. Coin tossing is a basic example of a random phenomenon: 2 2. ![]() Randomness is essential in statistics as well as in making a fair decision 1–4 1.
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