Probability ExamplesProbability Examples
Probability is ordinarily used to describe an attitude of mind towards some proposition
of whose truth we are not certain. The proposition of interest is usually of the form
"Will a specific event occur?" The attitude of mind is of the form "How certain are we
that the event will occur?"
The certainty we adopt can be described in terms of a numerical measure and this
number, between 0 and 1, we call probability. The higher the probability of an
event, the more certain we are that the event will occur.
Thus, probability in an applied sense is a measure of the confidence a person has
that a (random) event will occur.
The concept has been given an axiomatic mathematical derivation in probability theory,
which is used widely in such areas of study as mathematics, statistics, finance, gambling,
science, artificial intel igence/machine learning and philosophy to, for example, draw
inferences about the expected frequency of events. Probability theory is also used to
describe the underlying mechanics and regularities of complex systems.KnowMoreAboutExplainDifferentiation Tutorcircle.comPageNo.:1/4
The scientific study of probability is a modern development. Gambling shows that there
has been an interest in quantifying the ideas of probability for mil ennia, but exact
mathematical descriptions arose much later.
There are reasons of course, for the slow development of the mathematics of probability.
Whereas games of chance provided the impetus for the mathematical study of probability,
fundamental issues are stil obscured by the superstitions of gamblers.
According to Richard Jeffrey, "Before the middle of the seventeenth century, the term
'probable' (Latin probabilis) meant approvable, and was applied in that sense, univocal y,
to opinion and to action. A probable action or opinion was one such as sensible people
would undertake or hold, in the circumstances." However, in legal contexts especial y,
'probable' could also apply to propositions for which there was good evidence.
Aside from elementary work by Girolamo Cardano in the 16th century, the doctrine of
probabilities dates to the correspondence of Pierre de Fermat and Blaise Pascal (1654).
Christiaan Huygens (1657) gave the earliest known scientific treatment of the subject
Jakob Bernoul i's Ars Conjectandi (posthumous, 1713) and Abraham de Moivre's Doctrine
of Chances (1718) treated the subject as a branch of mathematics.
See Ian Hacking's The Emergence of Probability and James Franklin's The Science of
Conjecture for histories of the early development of the very concept of mathematical
In a deterministic universe, based on Newtonian concepts, there would be no probability if
al conditions are known, (Laplace's demon). In the case of a roulette wheel, if the force of
the hand and the period of that force are known, the number on which the bal wil stop
would be a certainty. ReadMoreAboutAnalyticalGeometryProblems Tutorcircle.comPageNo.:2/4
Of course, this also assumes knowledge of inertia and friction of the wheel, weight,
smoothness and roundness of the bal , variations in hand speed during the turning and so
forth. A probabilistic description can thus be more useful than Newtonian mechanics for
analyzing the pattern of outcomes of repeated rolls of roulette wheel.
Physicists face the same situation in kinetic theory of gases, where the system, while
deterministic in principle, is so complex (with the number of molecules typically the order
of magnitude of Avogadro constant 6.02*1023) that only statistical description of its
properties is feasible.
Probability theory is required to describe nature. A revolutionary discovery of early
20th century physics was the random character of al physical processes that occur at
sub-atomic scales and are governed by the laws of quantum mechanics.
The objective wave function evolves deterministically but, according to the Copenhagen
interpretation, it deals with probabilities of observing, the outcome being explained by a
wave function collapse when an observation is made.
However, the loss of determinism for the sake of instrumentalism did not meet with
universal approval. Albert Einstein famously remarked in a letter to Max Born: "I am
convinced that God does not play dice".
Like Einstein, Erwin Schrodinger, who discovered the wave function, believed quantum
mechanics is a statistical approximation of an underlying deterministic reality. In modern
interpretations, quantum decoherence accounts for subjectively probabilistic behavior. TutTu ot rcr ic rcr lc el .e c. oc mPaP geg e NoN ..::2/3 3/4