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Summary Introduction to The Probability Theory.
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- University Of South Africa (Unisa)
A masterclass by Makotsi Emmanuel.
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THE MATHEMATICS OF UNCERTAINITY(PROBABILITY)
“WRITTEN BY EMMANUEL MAKOTSI”
The purpose of these study notes is to introduce you to some
elementary aspects of the probability theory.
Introduction.
The world we live in is defined by uncertainties.
No one knows when they are going to die.
No one knows with certainty which horse is going to win a race.
More so, it is not even possible for one to predict with certainty
how they will be feeling tomorrow.
The idea of probability according to Mathematical history arises
in 1654, when Chevalier de Mere, a wealthy Frenchman with a
habit of gambling proposed to Blaise Pascal, a Mathematician
of the 17th century how the prize money of a simple game of
chance would be divided between two players if the game is
terminated before either one player wins.
Pascal then communicated the question with another famous
mathematician of that time, Pier de Fermat.
Both Pascal and Fermat arrived at the same answer to the
problem however their methods of solution were different.
The answers were based on the probability or likelihood each
player had on winning the game.
From this idea, a new field of Mathematics emerged.
By 1713, Jacob Bernoulli anticipated by more than two
centuries, the profound practical influence of the subject by
suggesting that probability might be applied the question of
Government, Law, Economics, and morality.
Essential Definitions
Definition 1
, Probability can be divided as the number of times an event A,
of interest occurs over the total number of times of experiment
was conducted.
Mathematically, this is represented as follows: -
P[A] = number of times A occurs
Total number of times experiment was done
Definitions 2.
A statistical experiment is a random experiment whose
outcome are uncertain.
An example of an experiment is tossing a coin or rolling a die.
Basic properties of probability
1. If an event A cannot occur, then its probability of
occurrence is zero. Mathematically written as: -
P [A] = 0
2. If the event A is certain to happen, then probability of A
occurring is one, given as: -
P [A] = 1
3. A probability value will always lie in the interval zero to one
indicated by the expression: -
0 ≤ P [A] ≤ 1
The higher the probability, i.e. the closer to one the
probability, the more likely the event is to happen.
The lower the probability, thus the less likely the event to
happen.
4. The probability of A not occurring is given by 1minus the
probability of A occurring, Mathematically
P [A] =1 - P [A]
There are a lot if approaches to probability which include: -
the experimental approach
theoretical approach
Experimental approach
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