Lecture 13

A deterministic model is defined as a model which stipulates that the conditions under which an experiment is performed determine the outcome of the
experiment For a number o.f situations the deterministic model suffices. However,
there are phenomena [as covered by (U) above] which do not leml themselves to
deterministic approach and are known as unpredictable or probabilistic
phenomena. For example
(i) In tossing o.f ac o.in one is not sure if a head o.r tail will be obtained.
(Ii) If a light tube has lasted for I hours, no.thing can be said out its further
life. It may fail to. function any moment
In such cases we talk o.f chance or probability which is taken tobe a quantitative
measure o.f certainty.
4·2. Short· History. Galileo (1564-1642), an Italian mathematic~, was the
first to attempt at a quantitative measure of probability while dealing with some
problems related to the theory o.f dice in gambiing. But the flfSt fo.undatio.n o.f the
mathematical theory ifprobabilily was laid in the mid-seventeenth century by lwo.
French mathematicians, B. Pascal (1623-1662) andP. Fermat (1601-1665), while.
4·2 Fundamentals of Mathematical Statistics I
solving a number of problems posed by French gambler and noble man Chevalier-
De-Mere to Pascal. The famous problem of points posed by De-Mere to Pascal
is : "Two persons playa game of chance. The person who ftrst gains a certain
number of pOints wins the stake. They stop playing before the game is completed.
How is,th~ stake to be decided on the.basis of the number of points each has won?"
The two mathematicians after a lengthy correspondence between themselves
ultimately solved this problem and dfis correspondence laid the fust foundation of
the science of probability. Next stalwart in this fteld was J. . Bernoulli (1654-1705)
whose Treatise on Probability was publiShed posthwnously by his nephew N.
Bernoulli in 1719. De"Moivre (1667-1754) also did considerable work in this field
and published his famous Doctrine of Chances in 1718. Other.:main contributors
ai e: T. Bayes (Inverse probability),P.S. Laplace (1749-1827) who after extensive
research over a number of ·years ftnally published Theoric analytique des prob-
abilities in 1812. In addition 10 these, other outstanding cOntributors are Levy,
Mises and R.A. Fisher.
Russian mathematicians also have made very valuable contribUtions to the
modem theory of probability. Chief contributors, to mention only a few of them
are,: Chebyshev (1821-94) who foUnded the Russian School of Statisticians;
A. Markoff (1856-1922); Liapounoff (Central Limit Theorem); A. Khintchine
(law of Large Numbers) and A. Kolmogorov, who axibmi.sed the calculus of
probability. .
4·3. DefinitionsorVariousTerms. In this section we wiUdefane and explain
the various tenns which are used in the definition of probability.
Trial and Event. Consider an experiment which, though repeated under
essentially identical conditions, does not give unique results but may result in any
one of the several possible outcomes.The experiment is known as a trial and the
outcomes are known as events or casts. For example:
(i) Throwing of a die is a trial and getting l(or 2 or 3, ... or 6) i~ an event
(ii) Tossing of a coin is a trial and getting head (H) or tail (T) is an event
(iii) Omwing two cards from a pack of well-shuffled· cards is a.trial.
Exhaustiye Events. The !ptal number of possible outcomes in ~y trial ~
known il$ exhaustive events or ~ltaustive cases. For example :
(;) II) tossing of a coin there are two exhaustive cases, viz., head and. tai1[
(the possibiljty of the co~ ~tanding on an edge being ignOred).
(ii) In throwing of a die, there are six, exhaustive cases since anyone oft he
6 faces 1,2, ... ,6 may come uppermost.
(iii) In draw!ng two cards from a pack of cards the ~x~.austive nwnberO(
cases is lCZ, since 2 cards can be dmwn out of 52 cards in ,zCz ways. ~
(iv) In tlu 9wing of ~o dice, the exhaustive number of cases is 62 =3 6,
since any of the 6 numbers 1.1.Q 6 on. the fustdie.can be associated with any oftbr
six numbers on th~ other die.
Theory or l i-obabllity