Probability and its usesEvangelia Antonaki

Maria Chantzopoulou

Konstantina Theologi

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Chapters

•Leonhard Euler

•Jean le Rond d’Alembert

•Poisson

•Buffon and the Needle Problem

•Bernoulli and Smallpox

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Chapter 1 Buffon and the needle problem

Georges-Luis Leclerc, Comte de Buffon (7 September 1707- 16 April 1788) was a French naturalist, mathematician, cosmologist and encyclopedic author. His works influenced the next two generations of naturalists.

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He treats in detail the famous “Needle Problem”“Suppose a needle is thrown at random on a floor marked with equidistant parallel lines. What is the probability that the needle will land on one of the lines?”

To solve this problem, we need to idealize the physical objects by assuming that the floorboards have a uniform width. Let L be the length of the needle and d be the width of the floorboard. We willalso assume that the needle has length L<d so that the needle cannot cross more than one crack. Finally, we assume that the cracks between the floorboards and the needle are line segments.

We can furthermore assume that the lines are horizontal and so we only need to consider what can happen when the needle lands between a single “strips” as indicated in the next picture.

 

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Here, y is the distance from the lowest point of the needle to the nearest line above it. If the needle happens to fall horizontally, then y is simply the distance from the needle to the nearest line above it. So 0< y <d. Let θ represent the angle between the needle and one of the parallel lines (preferably the line above) so that 0 <θ

<π . Then the ordered pair (θ, y) uniquely determines the position of the needle up to vertical translations by integer multiples of d,

and by any horizontal translation. So the square: {(θ, y) l 0 ≤ θ < π, 0 ≤ y < d}

Forms the sample space. The quantity y=L sin (θ) is then the vertical height of the needle. Now the needle will intersect one of

the lines if and only if y< Lsin(θ).

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Buffon’s needle refers to a simple estimation of the value of pi. You will notice that if you drop the needle on a table or on the floor one of the two things happens: (1) The needle crosses or touches one of the lines, or (2) the needle crosses no lines. The idea now is to keep dropping this needle over and over on the table (or the floor), and to record the statistics. Namely, we want to keep track of both the total number of times that the needle is randomly dropped on the table (call this N), and the number of times that it crosses a line (call this C). If you keep dropping the needle, eventually you will find that the number 2N/C approaches the value of pi!

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Chapter 2

Daniel Bernoulli and Smallpox

Daniel Bernoulli (8 February 1700 –8 March 1782) was a Dutch-Swiss mathematician and was one of the many prominent mathematicians in the Bernoulli family. He is particularly remembered for his applications of mathematics to mechanics, especially fluid mechanics, and for his pioneering work in probability and statistics.

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One of the earliest attempts to analyze a statistical probleminvolving censored data was Bernoulli's 1766 analysis of smallpox morbidity and mortality data to demonstrate theefficacy of vaccination.In 1760 he submitted to the Academy of sciences in Paris awork entitled an attempt at the new analysis of the mortality caused by smallpox and of the advantages of inoculation to prevent it. The question was whether inoculation (the voluntary introduction of a small amount of less virulent smallpox in the body to protect it against later infections) should be encouraged even if it is sometimes a deadly operation. This technique has been known for a long time in Asia and England, but in Franceinoculation was considered reluctantly.

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Ιn 1759, Daniel Bernoulli studied the inoculation problem from a mathematical point of view. More precisely, the challenge was to find a way of comparing the long-term benefit of inoculation with the immediate risk of dying. For this purpose, Bernoulli made the following simplifying assumptions:  People infected with smallpox for the first time die with a probability P (independent of age) and survive with a probability 1 – p. Everybody has a probability q of being infected each year ; more precisely , the probability for one individual to become infected between age x and age x + dx is qdx, where dx is an infinitesimal time period .People surviving from smallpox are protected against new infections for the rest of their life ( they have been immunized ) .

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Daniel Bernoulli was, by a long way, the first to express the

proportion of susceptible individuals of anendemic infection in terms of the force of infection and life expectancy. His formula is valid for arbitrary age dependent host mortality, in contrast to some current formulas which underestimate herd immunity. Therefore, it is more accurate to use the more general formula derived by Bernoulli.

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Chapter 3 Leonhard Euler

He was a Swiss mathematician perhaps one of the most important people in the history of mathematics. .He was one of the most active mathematicians that ever appeared. His colleagues used to call him “The Analysis Incarnate”.

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He studied probabilities in games of chance. He is rather popular about his study in the Genoese lottery. In “Genoese Lottery” participants would bet on the drawing of five balls from a wheel, which contained balls numbered 1, 2, 3, ¼, and 90. He calculated the bank's profit using the formula:

 Where E is the expected value and P is price of a ticket. Although Euler’s work in the theory of probability extended our understanding about games of chance, but he did not invent new applications of the theory.

P-E E ×100%

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Chapter 4

Jean le Rond D’Alembert (1717 –1783) He was a French mathematician and philosopher who acted in the period of Enlightenment. At some time in his life, d’Alembert had a dispute with the Swiss mathematician Daniel Bernoulli about the prevention of smallpox.

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At their time people could be protected from smallpox with a process called variolation. D’Alembert disagreed with Bernoulli that people should be variolated from a young age. Then he gave an example of a 30 year old man. According to Bernoulli’s theory, he insisted, variolation would let the man live approximately till the age of 57. The risk of dying of variolation was estimated at 1/200, and it is the 1/200 chance of almost immediate death that should be of more concern to the 30-year-old than the possibility of adding a few years to the end of what was, for the time, a long life. D’Alembert believed that it would be wiser for the man to enjoy his life, rather than risk it.

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Chapter 2

Poisson Siméon-Denis Poisson, (1781, 1840), French mathematician known for his work on definite integrals, electromagnetic theory, and probability. He published between 300 and 400 mathematical works in all.

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The distribution equationIf the expected number of occurrences in a given interval is λ, then the probability that there are exactly k occurrences (k being a non-negative integer, k = 0, 1, 2,...) is equal to:

Where:e is the base of the natural logarithm(e = 2.71828...)k is the number of occurrences of an event — the probability of which is given by the functionk! is the factorıalof kλ is a positive real number, equal to the expacted number of occurrences during the given interval.

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Probability distribution function of Poisson:• for λ= 1• for λ = 5 • for λ = 10

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Assumptions of Poisson

(a) The probability of an event to happen in a short time is proportional to the amount of space.(b) In a short time, the probability of two events occurs is almost nil.(c) The likelihood of a number of events in a given period is independent from the starting point of space.(d) The likelihood of a number of events in a given period is independent of the number of events that occurred before that time. 

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OccurrenceThe Poisson distribution arises in connection with Poisson processes. It applies to various phenomena of discrete properties (that is, those that may happen 0, 1, 2, 3,... times during a given period of time or in a given area) whenever the probability of the phenomenon happening is constant in time or space.

Examples of events that may be modeled as a Poisson distribution include:• Number of telephone calls arriving at a call center in a given

period.• Number of customers visiting a store in a period of time or

number of customers of a store who buy a particular product during the rebate etc.

• Number of individuals in a population who live more than 90 years.