Some theoretical probabilities of matching birthdays for n people where
\\n\\n
Released this past November, the list is based on data collected from the Web of Science and highlights some of the world’s most influential scientific minds by naming the researchers whose publications over the previous decade have included a high number of Highly Cited Papers placing them among the top 1% most-cited.
\\n\\nWe wish to congratulate all of the researchers named and especially our authors on this amazing accomplishment! We are happy and proud to share in their success!
\\n"}]',published:!0,mainMedia:null},components:[{type:"htmlEditorComponent",content:'IntechOpen is proud to announce that 179 of our authors have made the Clarivate™ Highly Cited Researchers List for 2020, ranking them among the top 1% most-cited.
\n\nThroughout the years, the list has named a total of 252 IntechOpen authors as Highly Cited. Of those researchers, 69 have been featured on the list multiple times.
\n\n\n\nReleased this past November, the list is based on data collected from the Web of Science and highlights some of the world’s most influential scientific minds by naming the researchers whose publications over the previous decade have included a high number of Highly Cited Papers placing them among the top 1% most-cited.
\n\nWe wish to congratulate all of the researchers named and especially our authors on this amazing accomplishment! We are happy and proud to share in their success!
\n'}],latestNews:[{slug:"stanford-university-identifies-top-2-scientists-over-1-000-are-intechopen-authors-and-editors-20210122",title:"Stanford University Identifies Top 2% Scientists, Over 1,000 are IntechOpen Authors and Editors"},{slug:"intechopen-authors-included-in-the-highly-cited-researchers-list-for-2020-20210121",title:"IntechOpen Authors Included in the Highly Cited Researchers List for 2020"},{slug:"intechopen-maintains-position-as-the-world-s-largest-oa-book-publisher-20201218",title:"IntechOpen Maintains Position as the World’s Largest OA Book Publisher"},{slug:"all-intechopen-books-available-on-perlego-20201215",title:"All IntechOpen Books Available on Perlego"},{slug:"oiv-awards-recognizes-intechopen-s-editors-20201127",title:"OIV Awards Recognizes IntechOpen's Editors"},{slug:"intechopen-joins-crossref-s-initiative-for-open-abstracts-i4oa-to-boost-the-discovery-of-research-20201005",title:"IntechOpen joins Crossref's Initiative for Open Abstracts (I4OA) to Boost the Discovery of Research"},{slug:"intechopen-hits-milestone-5-000-open-access-books-published-20200908",title:"IntechOpen hits milestone: 5,000 Open Access books published!"},{slug:"intechopen-books-hosted-on-the-mathworks-book-program-20200819",title:"IntechOpen Books Hosted on the MathWorks Book Program"}]},book:{item:{type:"book",id:"9413",leadTitle:null,fullTitle:"Essentials in Hip and Ankle",title:"Essentials in Hip and Ankle",subtitle:null,reviewType:"peer-reviewed",abstract:"Each chapter of this book covers physical examination, imaging, differential diagnoses, and treatment. 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M. Sohel",middleName:null,surname:"Murshed",slug:"s.-m.-sohel-murshed",fullName:"S. M. Sohel Murshed"}],productType:{id:"1",title:"Edited Volume",chapterContentType:"chapter",authoredCaption:"Edited by"}}},ofsBook:{item:{type:"book",id:"8176",leadTitle:null,title:"DNA Methylation Mechanism",subtitle:null,reviewType:"peer-reviewed",abstract:"‘Simplicity is the ultimate sophistication’- Leonardo Da Vinci“Thus, joining the rigor of the demonstrations of science to the uncertainty of fate, and reconciling these two seemingly contradictory things, it can, taking its name from both, appropriately arrogate to itself this astonishing title: the geometry of chance.”
Blaise Pascal
“You believe in the God who plays dice, and I in complete law and order.”
Albert Einstein, Letter to Max Born
“Chance is the pseudonym of God when He did not want to sign.”
Anatole France
“There is a certain Eternal Law, to wit, Reason, existing in the mind of God and governing the whole universe.”
Saint Thomas Aquinas
“An equation has no meaning for me unless it expresses a thought of God.”
Srinivasa Ramanujan
Calculating probabilities is the crucial task of classical probability theory. Adding supplementary dimensions to nondeterministic experiments will yield a deterministic expression of the theory of probability. This is the novel and original idea at the foundations of my complex probability paradigm. As a matter of fact, probability theory is a stochastic system of axioms in its essence; that means that the phenomena outputs are due to randomness and chance. Adding new imaginary dimensions to the nondeterministic phenomenon happening in the set
The advantages and the purpose of the present chapter are to [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39]:
Extend the theory of classical probability to cover the complex numbers set, hence to connect the probability theory to the field of complex analysis and variables. This task was initiated and developed in my earlier 14 works.
Apply the novel paradigm and its original probability axioms to Monte Carlo techniques.
Prove that all phenomena that are nondeterministic can be transformed to deterministic phenomena in the complex probabilities set which is
Compute and quantify both the chaotic factor and the degree of our knowledge of Monte Carlo procedures.
Represent and show the graphs of the functions and parameters of the innovative model related to Monte Carlo algorithms.
Demonstrate that the classical probability concept is permanently equal to 1 in the set of complex probabilities; thus, no chaos, no randomness, no ignorance, no uncertainty, no unpredictability, no nondeterminism, and no disorder exist in
Prepare to apply this inventive paradigm to other topics in prognostics and to the field of stochastic processes. These will be the goals of my future research publications.
Regarding some applications of the novel established model and as a subsequent work, it can be applied to any nondeterministic experiments using Monte Carlo algorithms whether in the continuous or in the discrete cases.
Moreover, compared with existing literature, the major contribution of the current chapter is to apply the innovative complex probability paradigm to the techniques and concepts of the probabilistic Monte Carlo simulations and algorithms.
The next figure displays the major aims and purposes of the complex probability paradigm (CPP) (Figure 1).
The diagram of the major aims of the complex probability paradigm.
The simplicity of Kolmogorov’s system of axioms may be surprising [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14]. Let E be a collection of elements {E1, E2, …} called elementary events and let F be a set of subsets of E called random events. The five axioms for a finite set E are:
Axiom 1:F is a field of sets.
Axiom 2:F contains the set E.
Axiom 3: A nonnegative real number Prob(A), called the probability of A, is assigned to each set A in F. We have always 0 ≤ Prob(A) ≤ 1.
Axiom 4:Prob(E) equals 1.
Axiom 5: If A and B have no elements in common, the number assigned to their union is
hence, we say that A and B are disjoint; otherwise, we have
And we say also that
Moreover, we can generalize and say that for N disjoint (mutually exclusive) events
And we say also that for N independent events
Now, we can add to this system of axioms an imaginary part such that:
Axiom 6: Let
Axiom 7: We construct the complex number or vector
Axiom 8: Let Pc denotes the probability of an event in the complex probability universe
We can see that by taking into consideration the set of imaginary probabilities, we added three new and original axioms, and consequently the system of axioms defined by Kolmogorov was hence expanded to encompass the set of imaginary numbers.
To summarize the novel paradigm, we state that in the real probability universe
The EKA or the CPP diagram.
Let
The relative error in the Monte Carlo methods is
Additionally, the percent relative error is =
Moreover, we define the real probability in the set
= 1 − the relative error in the Monte Carlo method.
= probability of Monte Carlo method convergence in
And therefore,
= probability of Monte Carlo method divergence in the imaginary complementary probability set
Consequently,
= the relative error in the Monte Carlo method.
= probability of Monte Carlo method divergence in
In the case where
And in the case where
Consequently, if
And if
We have
where
and
That means that the complex random vector Z is the sum in
If
and
therefore
If
and
therefore
If
and
therefore
We have
From CPP we have that
then solving the second-degree equations for
That means that DOK is minimum when the approximate result
In addition, if
That means that DOK is maximum when the approximate result
We have
Since
From CPP we have that
and vice versa.
That means that Chf is minimum when the approximate result
In addition, if
And, conversely, if
That means that Chf is equal to 0 when the approximate result
We have
Since
From CPP we have that
and vice versa.
That means that MChf is maximum when the approximate result
In addition, if
And, conversely, if
That means that MChf is minimum and is equal to 0 when the approximate result
We have
Since
Then
Therefore,
That means that the slope of the probability of convergence in
That means that the slopes of the probabilities of divergence in
Additionally,
that means that the module of the slope of the complex probability vector Z in
Furthermore, since
which means that Pc is constantly equal to 1 for every value of
We can infer from what has been developed earlier the following:
The real probability of convergence
We have
The imaginary complementary probability of divergence
The real complementary probability of divergence
The random vector of complex probability
The degree of our knowledge
The Chaotic factor
The magnitude of the chaotic factor MChf
At any iteration number N
then
Thus, the prediction in the set
Let us consider afterward a multidimensional integral and a stochastic experiment to simulate the Monte Carlo procedures and to quantify, to draw, as well as to visualize all the prognostic and CPP parameters.
The flowchart that follows illustrates all the procedures of the elaborated prognostic model of CPP.
Note that all the numerical values found in the simulations of the new paradigm for any iteration cycles N were computed using the 64-bit MATLAB version 2020 software and compared to the values found by Microsoft Visual C++ programs. Additionally, the reader should be careful of the truncation and rounding errors since we represent all numerical values by at most five significant digits and since we are using Monte Carlo techniques of simulation and integration which yield approximate results under the influence of stochastic aspects and variations. We have considered for this purpose a high-capacity computer system: a workstation computer with parallel microprocessors, a 64-bit operating system, and a 64-GB RAM.
The Monte Carlo technique of integration can be summarized by the following equation:
Let us consider here the multidimensional integral of the following function:
with
Furthermore, the four figures (Figures 3–6) illustrate and prove the increasing convergence of Monte Carlo simulation and technique to the exact result
The increasing convergence of the Monte Carlo method up to N = 50 iterations.
The increasing convergence of the Monte Carlo method up to N = 100 iterations.
The increasing convergence of the Monte Carlo method up to N = 500 iterations.
The increasing convergence of the Monte Carlo method up to N = 100,000 iterations.
Moreover, Figure 7 shows undoubtedly and graphically the relation of all the parameters of the complex probability paradigm (
The CPP parameters and the Monte Carlo method for a multiple integral.
An interesting problem that can be solved using simulation is the famous birthday problem. Suppose that in a room of n persons, each of the 365 days of the year (not a leap year) is equally likely to be someone’s birthday. It can be proved from the theory of probability and contrary to intuition that only 23 persons need to be present for the probability to be better than fifty-fifty that at least two of them will have the same birthday.
Many people are interested in checking the theoretical proof of this statement, so we will demonstrate it briefly before doing the problem simulation. After someone is asked about his or her birthday, the probability that the next person asked will not have the same birthday is 364/365. The probability that the third person’s birthday will not match those of the first two people are 363/365. It is well-known that the probability of two independent and successive events happening is the product of the probability of the separate events. In general, the probability that the nth person asked will have a birthday different from that of anyone already asked is
The probability that the nth person asked will provide a match is 1 minus this value:
which shows that with 23 persons, the chances are 50.7%; with 55 persons, the chances are 98.6% or almost theoretically certain that at least two out of 55 people will have the same birthday. The table gives the theoretical probabilities of matching birthdays for a selected number of people n (Table 1).
Number of people n | Theoretical probability = |
---|---|
n = 5 | P = 0.027135573700 |
n = 10 | P = 0.116948177711 |
n = 15 | P = 0.252901319764 |
n = 20 | P = 0.411438383581 |
n = 22 | P = 0.475695307663 |
n = 23 | P = 0.507297234324 |
n = 25 | P = 0.568699703969 |
n = 30 | P = 0.706316242719 |
n = 35 | P = 0.814383238875 |
n = 40 | P = 0.891231809818 |
n = 45 | P = 0.940975899466 |
n = 50 | P = 0.970373579578 |
n = 55 | P = 0.986262288816 |
n = 100 | P = 0.999999692751 |
n = 133 | P = 0.999999999999 |
n = 365 | P = 1.000000000000 |
Some theoretical probabilities of matching birthdays for n people where
Without using the probability theory, we can write a routine that uses the random number generator to compute the approximate chances for groups of n persons. Obviously, what is needed here is to choose n random integers from the set of integers {1, 2, 3, …, 365} and to check whether there is a match. When we repeat this experiment a large number of times, we can calculate afterward the probability of at least one match in any gathering of n persons. Note that if
Furthermore, the four figures (Figures 8–11) illustrate and prove the increasing convergence of Monte Carlo simulation and technique to the exact result
The increasing convergence of the Monte Carlo method up to N = 50 iterations.
The increasing convergence of the Monte Carlo method up to N = 100 iterations.
The increasing convergence of the Monte Carlo method up to N = 500 iterations.
The increasing convergence of the Monte Carlo method up to N = 750,000 iterations.
Moreover, Figure 12 shows undoubtedly and graphically the relation of all the parameters of the complex probability paradigm (
The CPP parameters and the Monte Carlo techniques for the matching birthday problem.
In Figure 13 and in the first cube, the simulation of Chf and DOK as functions of the iterations N and of each other is executed for the problem of matching birthday. If we project Pc2(N) = DOK(N) − Chf(N) = 1 = Pc(N) on the plane N = 0 iterations, we will get the line in cyan. The starting point of this line is point J (DOK = 1, Chf = 0) when N = 0 iterations, and then the line gets to point (DOK = 0.5, Chf = −0.5) when N = 375,000 iterations and joins finally and again point J (DOK = 1, Chf = 0) when N = NC = 750,000 iterations. The graphs of Chf(N) (pink, green, blue) in different planes and DOK(N) (red) represent the other curves. We can notice that point K (DOK = 0.5, Chf = −0.5, N = 375,000 iterations) is the minimum of all these curves. We can notice also that point L has the coordinates (DOK = 1, Chf = 0, N = NC = 750,000 iterations). Additionally, the three points J, K, and L correspond to the same points that exist in Figure 12.
Chf and DOK in terms of each other and of N for the problem of matching birthday.
In Figure 14 and in the second cube, we simulate the probability of convergence Pr(N) and its complementary real probability of divergence Pm(N)/i as functions of the iterations N for the problem of matching birthday. If we project Pc2(N) = Pr(N) + Pm(N)/i = 1 = Pc(N) on the plane N = 0 iterations, we will get the line in cyan. The starting point of this line is point (Pr = 0, Pm/i = 1), and the final point is point (Pr = 1, Pm/i = 0). The graph of Pr(N) in the plane Pr(N) = Pm(N)/i is represented by the red curve. The starting point of this graph is point J (Pr = 0, Pm/i = 1, N = 0 iterations), and then it gets to point K (Pr = 0.5, Pm/i = 0.5, N = 375,000 iterations) and joins finally point L (Pr = 1, Pm/i = 0, N = NC = 750,000 iterations). The graph of Pm(N)/i in the plane Pr(N) + Pm(N)/i = 1 is represented by the blue curve. We can notice how much point K is important and which is the intersection of the blue and red graphs when Pr(N) = Pm(N)/i = 0.5 at N = 375,000 iterations. Additionally, the three points J, K, and L correspond to the same points that exist in Figure 12.
Pm/i and Pr in terms of each other and of N for the problem of matching birthday.
In Figure 15 and in the third cube, we simulate the vector of complex probabilities Z(N) in
The vector of complex probability Z in terms of N for the problem of matching birthday.
In the current chapter, the extended and original Kolmogorov model of eight axioms (EKA) was connected and applied to the random and classical Monte Carlo techniques. Thus, we have bonded Monte Carlo algorithms to the novel CPP paradigm. Accordingly, the paradigm of “complex probability” was more expanded beyond the scope of my 14 earlier studies on this topic.
Also, as it was proved and demonstrated in the original paradigm, when N = 0 (before the beginning of the random simulation) and when N = NC (after the convergence of Monte Carlo algorithm to the exact result), then the chaotic factor (Chf and MChf) is 0, and the degree of our knowledge (DOK) is 1 since the stochastic aspects and variations have either not commenced yet or they have terminated their job on the random phenomenon. During the course of the nondeterministic phenomenon (N > 0), we have 0 < MChf ≤ 0.5, 0.5 ≤ DOK < 1, and − 0.5 ≤ Chf < 0, and it can be noticed that throughout this entire process, we have continually and incessantly Pc2 = DOK − Chf = DOK + MChf = 1 = Pc, which means that the simulation which seemed to be random and nondeterministic in the set
Moreover, it is important to mention here that one essential and very well-known probability distribution was taken into consideration in the current chapter which is the uniform and discrete probability distribution as well as a specific generator of uniform random numbers, knowing that the original CPP model can be applied to any generator of uniform random numbers that exists in literature. This will yield certainly analogous results and conclusions and will confirm without any doubt the success of my innovative theory.
As a prospective and future challenges and research, we intend to more develop the novel conceived prognostic paradigm and to apply it to a diverse set of nondeterministic events like for other stochastic phenomena as in the classical theory of probability and in stochastic processes. Additionally, we will implement CPP to the first-order reliability method (FORM) in the field of prognostic in engineering and also to the problems of random walk which have huge consequences when applied to economics, to chemistry, to physics, and to pure and applied mathematics.
R | the events real set |
M | the events imaginary set |
C | the events complex set |
i | the imaginary number with i2=−1 or i=−1 |
EKA | extended Kolmogorov axioms |
CPP | complex probability paradigm |
Prob | any event probability |
Pr | the probability in the real set R = the probability of convergence in R |
Pm | the probability in the complementary imaginary set M that corresponds to the real probability set in R = the probability of divergence in M |
Pc | the probability in R of the event with its associated event in M = the probability in the set C=R+M of complex probabilities |
RE | the exact result of the random experiment |
RA | the approximate result of the random experiment |
Z | complex probability number = complex random vector = sum of Pr and Pm |
DOK = Z2 | the degree of our knowledge of the stochastic experiment or system, it is the square of the norm of Z |
Chf | |
MChf | the magnitude of the chaotic factor of Z |
N | the number of iterations cycles = number of random vectors |
NC | the number of iterations cycles till the convergence of Monte Carlo method to RE = the number of random vectors till convergence. |
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