Probability

The probability of a given event is an expression of possibility or chance of occurrence of an event. A probability is a number that ranges from o (zero) to 1 (one). Zero for an event that cannot occur and 1 for an event that is certain to occur. How the number is assigned would based on the interpretation of the word 'probability', there is no normal agreement about its interpretation and many people associate probability and chance with nebulous and mastic ideas. Though, broadly speaking there are four different schools of thought on the concept of probability.

Classical or a prior probability

The class is approach to probability is the simplest and oldest. It is originated in eighteenth century in problems pertaining to games of chance, like throwing of coins, deck of cards, or dice etc. The basic assumption underlying the classical theory is that the outcomes of a random experiment are ''equally likely'', the event whose probability is required consists of one or more possible outcomes of the given activity like when a die is rolled once, any of the six possible outcomes that is 1, 2, 3,4,5,6, can occur. These activities are referred to in the modern terminology as experiment which is a word that refers to the processes which result in various possible outcomes or observations. The word ''equally likely'', however undefined, convey the notion that each outcome of an experiment has the same chance of appearing as any other. So in a throw of a dice, the occurrence 1, 2, 3,4,5,6 are equally likely events.

The definition of probability is given by a French mathematician Laplace and is normally adopted by disciples of the classical schools are as follows:

Probability, it is said, is the ratio of the number of ''favorable'' cases to the total number of equally likely cases. If the probability of occurrence of a is represented by p (A). Then by this definition we have:

P (A) = number of favorable cases/ Total number of equally likely cases

For the calculation of probability we have to find out two things:

1. Number of favorable cases.

2. Total number of equally likely cases.

For e.g., if a coin is tossed, there are two equally likely answers a head or a tall. So the probability of a head is ½.

In the same way, if a dice is thrown, the probability of achieving an even number is 3/6 or ½ as three of the six equally possible results are even numbers.

Symbolically, if an event can happen in 'a' ways out if a total of 'n' equally likely & mutually exclusive ways then the probability of occurrence of the event (known its success) is represented by:

P= pr (A) = a/n

And the probability of non-occurrence of the event (known its failure) is given y:

Q = pr (Not A) or p (A) =     n - a/N     or b/n

= 1 - a/n or 1 - p or 1 = pr (A).

As the sum of the successful & unsuccessful outcomes is equal to the total number of event, we have

A + b =n

Dividing by n, a/N + b/n =1

So that p + q = 1.

The Probability, therefore, may be written as a ratio. The numerator of the fraction related to this ratio shows the number of successful for unsuccessful outcomes. While the denominator shows the total number of possible outcomes. The scale of probability expands from zero to unity (that is 1). When p = o, it represents impossibility of the event taking place, that is the event cannot takes place. Though, this is true only when the number of possible outcomes is limited. For e.g., the probability of throwing seven with a single dice is zero. On the other hand, when p = 1 it represents certainty, that is the event is bound to take place, in many cases, in practical life the probability lies between these two extremes 0 and 1.

The Classical probability is often known as prior probability as if we keep using orderly examples of unlaces dice, fair coin etc. We can state the result in advance (a priority) without tossing a coin, rolling a die, etc.

Illustration: - A bag contain 10 black and 20 white balls, a ball is to be pick at random. What is the probability that it is black?

Solution: - The total number of balls in the bag = 10 + 20 = 30

Number of black balls = 10

Probability of getting a black ball is

P (A) = number of favorable cases/Total number of equally-y likely cases or a/n

=10/30 = 1/3

Probability of not getting a black ball is

Q = 20/30 = 2/3

Thus, p + q    + 1/3 + 2/3 = 1.