Probability

properties of  Probability

It is usually difficult to explain to the general public what statisticians do .Many think of  As “math nerds”who seem to enjoy dealing  with numbers.

Theorem：

①For  each  event A:p（A）=1-p（A'）

②p（∮）=0

③If events A and B are such that A∈B,thenP（A）≤P（B）

④For each event A,P（A）≤1

⑤If A and B are any Two events,thenP（A∪B）=P（A）+P（B）-P（A∩B）.

⑥If A .B and C are any three events,then

P（A∪B∪C）=P（A）+P（B）+P（C）-P（A∩B）-P（A∩C）-P（B∩C）+P（A∩B∩）

Methods  of  Enumeration

We  begin with a consideration of multiplication principle.

Theorem:

①each of the n! arrangements of n different objects is called A  permutation of the n objects

②each of the nPr arrangements is called A permutation of n objects taken r at O time

③If r object are  selected from A set of n objects,and If the order of selection is noted,then the selected set of r objects  is called an ordered sample of size r.

④Sampling with  replacement occurs when an object is selected and then replaced before the next object is selected.

⑤Sampling without replacement occurs when an object is not replaced after it has been selected

⑥each of the nCr unordered subsets is called a combination of n objects taken r at a time ,where  nCr=（n r）=n!╱r!（n-r）!

⑦each of the nCr permutation of n objects ,r of one type and n-r of another type ,is called a distinguishable permutation

Conditional Probability

Introduction the idea of conditional Probability by means of an example.

①The conditional probability of an event A,given that event B has occurred,is defined by P（AΙB）=P（A∩B）╱P（B）,provided that P（B）>0

②The probability that two events ,A and B , both occur is given by the multiplication rule,P（A∩B）=P（A）P（BΙA）,provideP（A）>0 or by P（A∩B）=P（B）P（AΙB） provided P（B）>0

                                  -  RobertV.Hogg