Theory of Probability

                Probability and chance are concepts we all understand We talk about how probable it is that a bus will arrive on time, or how likely it is that wa will get a passing grade in a course. These statements represent a degree of uncertainty in how we anticipate certain future events.
          Uncertainties are ever present in everyone's life. However in order to express the degree of uncertainty in numerical terms, rather than by using some vague, loose expression, we need a knowledge of the rules and operations of probabilities.
          In a statistics course, probability is even more important than in everyday life because it is linked to the concept of random sampling. We use proba-bilities to construct models to describe the likelihood of whether certain outcomes will occur in a random sample taken from a population. The
results of such a model can be displayed in a tabular format similar to that of frequency table. Relative frequencies and probabilities have a similar interpretation.
      Therefore it becomes necessary to deal with some of the basic laws of probability We can approach the concept of probability using

 (1)  The Classical Approach
 (2)  The Relative Frequency Approach
 (3)  Subjective-probability approach

 Each of these approaches has its  place in the overall study of probability, and no one point of view can handle all cases. A simple example of each will make the distinctions among them clear for our purposes

CLASSICAL APPROACH
When we roll a balanced die, any one of six sides may turn up It is customary when using the classical approach to assign probabilitias of 1/6 (1divide by 6) to each side, on the assumption that each side has an equal chance of turning up.

RELATIVE-FREQUENCY APPROACH
If our die is not symmetrical, or if it is loaded, it is illogical to assume that each side has an equal chance of being turned up. One.way to assess the probabilities associated with each side is to roll the die a great number of times and observe the relative frequency with which each side appears. This
might be called an empirical approach to assigning probabilities.

SUBJECTIVE-PROBABILITY APPROACH
Consider the case of a person who is trying to decide whether or not to change jots The person is concerned about the probability of succending in a new job. However, it would be futile for him or her to use the classical approach and assign the probability of success as 1/2 and the probability of
failure as 1/2. Nor rould one use the relative-frequency approach, repeating the process of quitting a great number of times in order to get an empirical feel for the probability of success. The person can. however, pull together at the information available and make a subjective judgment of the probability
of success.
In this unit we shall usually assign probabilities according to the classical approach, because random sampling validates the assumption that the possible outcomes of the sampling are equally likely. In addition, we shall present several rules and theorems that apply regardless of the approach
used to assign probabilities.
Much of what follows is presented in illustrations of games of chance, or in terms of samples drawn from an um. Though these illustrations make it easier to conceptualize the ideas presented, you should keep in mind tha probability principles have uses that range far beyond raffles and gaming
tables.

VOCABULARY
Addition rule                                                               Mutually exclusive events
Combination rule                                                        Ordered sample space
Complement of an event                                             Permutation
Compound event                                                         Probability
Conditional probability                                               Sample space
Dependent events                                                        Sampling with replacement
Equally likelihood                                                       Sampling without replacement
Event                                                                           Simple event
Experiment                                                                  Tree diagram
Factorial                                                                      +47Trial
Independent event                                                       Union of events
Intersection of events                                                  Unordered sample space
M-N rule                                                                      Urn model
Multiplication rule

 BASIC TERMINOLOGY

1. Random Experiment. If in each trial of an experiment conducted under identical conditions, the outcome is not unique, but may be any one of the possible outcomes, then such an experiment is called a random experiment.

Examples of random experiments are : tossing a coin, throwing a die, selecting a card from a pack of playing cards, selecting a family out of a given group of families, etc. In all these cases, there are a number of possible results which can occur but there is an uncertainty as to which one of them will actually occur.

Notes :
 (i) A die is a small cube used in gambling. On its six faces, dots are marked as
Plural of die is dice. The outcome of throwing a die is the number of dots on its uppermost face.

(ii) A pack of cards consists of four suits called Spades, Hearts, Diamonds and Clubs. Each suit
consists of 13 cards, of which nine cards are numbered from 2 to 10, an ace, king, a queen and a
jack (or knave). Spades and clubs are black-faced cards, while hearts and diamonds are red- faced cards.

2. Outcome: The result of a random experiment will be called an outcome.
.
3. Trial and Event: Any particular performance of a random experiment is called a trial and outcome or combination of outcomes are termed as events. For example;

(i) If a coin is tossed repeatedly, the result is not unique. We may get any of the  two faces, head or tail. Thus tossing of a coin is a random experiment or trial getting of a head or tail is an event.

(ii) In an experiment which consists of the throw of a six-faced die and observing
the number of points that appear, the possible outcomes are 1, 2, 3, 4, 5, 6
   In the same experiment, the possible events could also be stated as
'Odd number of points';  'Even number of points'; ' Getting a point greater than 4'; and so on.

Event is called simple if it corresponds to a single possible outcome of the experiment
otherwise it is known as a compound or composite event. Thus in tossing of a single die the event
of getting  6' is a simple event but the event of getting an even number is a composite event.

4. Exhaustive Events or Cases: The total number of possible outcomes of random experiment is known as the exhaustive events or cases. For example,
(i) In tossing of a coin, there are two exhaustive cases, viz., head and tail
(the possibility of the coin standing on an edge being ignored).

(ii) In throwing of a die, there are 6 exhaustive cases since any one of the 6 faces
1,2, ..., 6 may come uppermost.

(iii) In drawing two cards from a pack of cards, the exhaustive number of cases is
52C2, since 2 cards can be drawn out of 52 cards in 52C2, ways.

(iv) In throwing of two dice, the exhaustive number of cases is 62 = 36, since any of the numbers 1 to 6 on the first die can be associated with any of the 6 numbers on the other die. In general, in throwing of n dice, the exhaustive number of cases is 6".

5. Favourable Events or Cases: The number of cases favourable to an event in a trial is the number of outcomes which entail the happening of the event. For example,

(i) In drawing a card from a pack of cards the drawing of an ace is 4, for drawing a spade is 13 and for drawing a red card is 26.

(ii) In throwing of two dice, the number of cases favourable to getting the sum 5 is :
(1, 4), (4, 1), (2, 3), (3, 2), i.e., 4.

6. Mutually Exclusive Events: Events are said to be mutually exclusive or
 incompatible if the happening of any one of them precludes the happening of all the others, i.e., if no two or more of them can happen simultaneously in the same trial. For example,

(i) In throwing a die all the 6 faces numbered 1 to 6 are mutually exclusive since if any one of these faces comes, the possibility of others, in the same trial, is ruled out.

(ii) Similarly in tossing a coin the events head and tail are mutually exclusive.

7. Equally Likely Events:Outcomes of trial are said to be equally likely if taking into consideration all the relevant evidences, there is no reason to expect one in preference to the others.
 For example,
(i) In a random toss of an unbiased or uniform coin, head and tail are equally likely events.

(ii) In throwing an unbiased die, all the six faces are equally likely to come.

8. Independent Events: Several events are said to be independent if the happening
(or non-happening) of an event is not affected by the supplementary knowledge
concerning the occurrence of any number of the remaining events. For example,

(i) In tossing an unbiased coin, the event of getting a head in the first toss is
Independent of getting a head in the second, third and subsequent throws.

(ii) When a die is thrown twice, the result of the first throw does not affect the
result of the second throw.

(iii) If we draw a card from a pack of well-shuffled cards and replace it before
drawing the second card, the result of the second draw is independent of the first
draw. But, however, if the first card drawn is not replaced then the second draw is
dependent on the first draw.

Keep Reading
Keep Visiting
A R Statistics