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We carried out five experiments separately. Our experiments with perfectly thin coins enable us to consider only the statistical features of coin-tossing problems. To rule out physical and mechanical aspects of tossed coins, we used an online virtual coin toss simulation application ( ) with an ideal coin of zero thickness, where there is no bias between heads and tails, ensuring equal probabilities for heads and tails. In this study, we would like to assume the β exponents to be approximately −0.6 in coin tosses (with two equal outcomes) and last digits of prime numbers (with four equal outcomes) with respect to equally likely outcomes.įirst, we conducted an analysis with coin tossing, as shown in Fig. Consequently, the condition R/ N → 0 at N → ∞ explains why randomness is valid only for large numbers, which is known as the law of large numbers in probability theory. The statistical expectation of R/ N ∼ N β ( β < 0) implies that the frequency of each outcome should become equal (because R/ N → 0) as the total number of repetitions increases ( N → ∞). From R ∼ N α, R/ N should have a simple power-law relation R/ N ∼ N β, where β = α − 1 (note that β < 0 because α < 1). Additionally, the range of relative frequency ( R/ N) between equally likely outcomes is defined as R / N = ( f i m a x − f i m i n ), which is equivalent to ( n i m a x − n i m i n ) / N. Such a power-law scaling commonly appears in statistics and physics. This tendency can be described by a power-law scaling R ∼ N α, where 0 < α < 1. In statistics, it is well known that the range of frequency ( R) tends to be larger for a larger size of the sample ( N). The range of frequency ( R) is defined as the difference between the maximum frequency ( n i m a x ) and the minimum frequency ( n i m i n ), consequently described as R = ( n i m a x − n i m i n ). The relative frequency of an outcome ( f i) is calculated by dividing n i by the total number of repetitions ( N or equally the size of the sample). The frequency of each outcome ( n i) can vary complicatedly according to experiments and conditions. The distribution of last digits of prime numbers is another important topic in particular, it is unclear that the four last digits are random or evenly distributed when numbers are large enough. 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.
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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.