Transcript Probability and its uses

Probability and its uses
Evangelia Antonaki
Maria Chantzopoulou
Konstantina Theologi
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Chapters
• Buffon and the
Needle Problem
• Bernoulli and
Smallpox
• Leonhard Euler
• Jean le Rond
d’Alembert
• Poisson
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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 will
also 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 problem
involving censored data was Bernoulli's 1766 analysis of
smallpox morbidity and mortality data to demonstrate the
efficacy of vaccination.
In 1760 he submitted to the Academy of sciences in Paris a
work 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 France
inoculation 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:
P-E ×100%
E
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.
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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 equation
If 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 function
k! is the factorıalof k
λ 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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Occurrence
The 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.