4.2 Approaches to Probability
Introduction
Probability has a very old history, it was originated in the games of chance related to gambling. For instance, throwing of dice or coin and drawing cards from a pack. Jerome Cardan (1501~1576), an Italian mathematician was the first man to write a book on the subject “Book on Games of chance” which was published in 1663 after his death. The probability formulae and techniques were developed by Jacob Bernoulli(1654-1705), De Moivre (1667-1754), Thomas Bayes(1702-1761) and Joseph Lagrange(1736- 1813). Pierre Simon, Laplace in the nineteenth century unified all these early ideas and compiled the first general theory of probability.
In the beginning, the probability theory was successfully applied at the gambling tables. But after some time, it was applied in the solution of social, political, economic and business problems. In fact, it has become a part of our everyday lives. We face uncertainty in personal and management decisions and use probability theory. Probability constitutes the foundation of statistical theory.
There are mainly three approaches to probability
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Classical approach
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Empirical approach
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Axiomatic approach
Random Experiment
An experiment can be considered as a random experiment if all the possible outcomes are known in advance and none of the outcomes can be predicted with certainty. e.g. throwing a dice, tossing a coin etc.
Trial & Event
When a random experiment is performed, it is called a trial and outcome or combinations of outcomes are termed as events. For example
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When a coin is tossed repeatedly, the result is not unique. We may get any of the two faces; head or tail. Thus, throwing a coin is a random experiment and getting of a head or tail is an event.
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In the similar manner, when a dice is thrown, it is called a random experiment. Getting any of the faces 1, 2, 3, 4, 5 or 6 is an event. Getting an odd no. or an even no., getting no. greater than 3 or lower than five, these are called events.
3. Similarly,drawing of two balls from an urn containing ‘a red balls and‘b’white balls is a trial and getting of both red balls, or both white balls, or one red and one white ball are events.
Mutually Exclusive events or cases
Two or more events are considered as mutually exclusive if the happening of any one of them excludes the happening of all others in the same experiment. For example, in toss of a coin, the events ‘head’ and ‘tail’ are mutually exclusive because if head comes, we can’t get tail and if tail comes we can’t get head. Similarly, in the throw of a die, the six faces numbered 1, 2, 3, 4, 5 and 6 are mutually exclusive. Thus, events are said to be mutually exclusive of no two or more of them can happen simultaneously.
Classical/priori Probability
It is the oldest and simplest approach. Under this approach, there is no need to physically perform the experiment. The basic assumption is that the outcomes of a random experiment are equally likely. e.g. in a throw of a dice, occurrence of 1,2,3,4,5,6 are equally likely event.
If a random experiment results in N exhaustive, mutually exclusive and equally likely outcomes out of which mare favourable to the happening of an event X then the probability of occurrence of Xi.e.P(X) is given by
Example 1. A bag containing 10 green and 20 red balls. A ball is drawn at random. What is the probability that it is green.
Sol. Total number of balls in the bag=10+20=30 Number of green balls = 10
= 10 /30 or 1/3
Empirical Probability
The classical definition is difficult to apply as soon as we move from the field of coins, cards, dice and other games of chance. It may not explain the actual results in certain cases e.g if a coin is tossed 20 times , we may get 14 heads and 6 tails. The probability of head is thus 0.7 and tail is 0.3. However, if experiment is carried out large number of times, we should expect approximately equal number of heads and tails.
If an experiment is performed repeatedly under essentially homogeneous and identical conditions then the limiting value of the ratio of the number of times the event occurs to the number of trials, as the number of trials become indefinitely large is known as the probability of happening of the event.
Axiomatic Approach
The axiomatic Probability theory is an attempt at constructing a theory of probability which is free from inadequacies of both the classical and empirical approaches. It plays an important role in rendering a reasonable amount of comprehensibility and tractability to the understanding of chance phenomenon at-least in the initial stages of any scientific inquiry into their structure and composition where other approaches are less comprehensible and tractable.
Addition Law
The probability of occurrence of either event A or event B of two mutually exclusive events is equal to the sum of their individual probability.
Mathematically, we can represent as P(A U B) =P(A)+P(B)
Proof:- If an event A can happen in a1 ways and B in a2 ways then The number of ways in which either event can happen in a1+a2 ways. Total number of possibilities is n.
Then by definition , the probability of either the first or second event happening is
The theorem can be extended to three or more mutually exclusive events, thus P(A B C) =P(A)+P(B)+P(C)
Example 2. A deck of 52 cards, one card is drawn. What is the probability that it is either a king or a queen?
Sol. There is four kings and four queens in a pack of 52 cards.
The probability of drawing a card that is king= 4/ 52
The probability of drawing a card that is queen= 4/ 52
Since the events are mutually exclusive, the probability that the card drawn is either a king or
a queen= 4/52 + 4/52 = 2/13
If two events A & B are not mutually exclusive (joint events) then the addition law can be stated as follows
The probability of the occurrence of either event A or event B or both is equal to the probability that event A occurs, plus the probability that event B occurs minus the probability that both events occur. I can be shown as
Example3. The managing committee of Residents Welfare Association formed a sub-committee of 5 persons to look into electricity problem. Profiles of the 5 persons are
Male age 40
Male age 43
Female age 38
Female age 27
Male age 65
If a chairperson has to be selected from this, what is the probability that he would be either female or over 32 years.