Counting and Probability for AMC-8

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 x ways of doing something and y ways of doing another thing and we can not do both at the same time, then there are x+y ways to choose one of the actions.

In terms of set theory, if S_1,S_2,\cdots S_n are disjoint sets, then |\bigcup_i S_i|=\sum_i |S_i|.

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 a ways and the second in b ways, then the task can be performed in a\cdot b ways.

In terms of set theory, |S_1\times S_2\times\cdots\times S_n|=\prod_i |S_i|.

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?

Answer: 3\times 4=12.

Sample Space and Events

Consider an experiment that can produce various outcomes. The set of all possible outcomes is called the sample space (S) of the experiment. Elements of the power set P(S) 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 A,

P(A):=\frac{|A|}{|S|}.

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