The notion of probability arose from games of chance like lotteries, bingos, and roulette. Though probability has now become a sophisticated field of Mathematics involving the use of Measure Theory and Analysis, its basics can be learnt using principles of counting.
The Addition Principle
if we have ways of doing something and ways of doing another thing and we can not do both at the same time, then there are ways to choose one of the actions.
In terms of set theory, if are disjoint sets, then .
Example : On a typical saturday evening, Jill either watches a film or a play. This week, there are 4 movies on the charts and the theatre has 3 plays. In how many ways can Jill spend her evening?
Answer : 4+3=7.
The Multiplication Principle
Suppose that a task can be done in two parts. If the first part can be done in ways and the second in ways, then the task can be performed in ways.
In terms of set theory, .
Example : Suppose that all roads from city A to city C pass through city B. There are 3 paths from city A to city B and 4 paths from city B to city C. How many paths are there from city A to city C?
Sample Space and Events
Consider an experiment that can produce various outcomes. The set of all possible outcomes is called the sample space () of the experiment. Elements of the power set of the sample space are known as events.
If event A and event B never occur simultaneously on a single performance of an experiment, then they are called mutually exclusive events.
Classical definition of probability
Given an event ,