# 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.

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?

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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