actuary notes[1]

random event

1. the theory of probability replies on random experiments which result can not be certainly confirmed.although we are unable to get the necessary consequence of one experiment,all reusults perhaps appear in those experiment can be found.so every testing result will be collected to construct sample space which is infinite or inifite.
2. for example,there are six spitballs written six different integer between 1 and 6 in one box.if you take one out of those spitballs,it must be written one of six numbers,one possible number will appears in your spitball with averaged possiblity.
   let A is sample space,$A=1,2,3,4,5,6$,so every number be selected with the probability of $\frac{1}{6}$.
   the situation was changed ,one box became three boxes,every box aslo includes six spitball written one of six numbers from 1 to 6,you still take one of every box,now the sample space A={{1,1,1},{1,1,2},......}A=\{\{1,1,1\},\{1,1,2\},......\}A={{1,1,1},{1,1,2},......} include $6\times 6\times 6=216$
3. let the subset $A^{\prime }\subset A$,$A$ can be called as event(random event) such as $A_1^{\prime },A_2^{\prime },...$,if $A^{\prime }$ just only involves one sample,then we call $A^{\prime }$ as basic event,in the above example, A′={{1,2,3}}A'=\{\{1,2,3\}\}A′={{1,2,3}} can be called as basic event.
4. let us suppose the event(random event) A′={{1,2,3},{2,2,3},{3,2,3}}A'=\{\{1,2,3\},\{2,2,3\},\{3,2,3\}\}A′={{1,2,3},{2,2,3},{3,2,3}} , if you fetch out three spitballs written 1,3 and 2 respectively,then the event $A^{\prime }$ has took place, $A^{\prime }$ have never been appeared,furthermore three spitballs written the numbers 3,1 and 1 are took out of three boxes respectively,so $A^{\prime }$ never happen because of the fact {1,2,3}∉A′\{1,2,3\} \notin A'{1,2,3}∈/A′.
5. let A′′={{1,2,3},{2,2,3}}A''=\{\{1,2,3\},\{2,2,3\}\}A′′={{1,2,3},{2,2,3}},so $A^{\prime \prime }\subset A^{\prime }$.
6. let A′′={{1,2,3},{2,2,3},{3,2,3}}A''=\{\{1,2,3\},\{2,2,3\},\{3,2,3\}\}A′′={{1,2,3},{2,2,3},{3,2,3}} and A′={{1,2,3},{2,2,3},{3,2,3}}A'=\{\{1,2,3\},\{2,2,3\},\{3,2,3\}\}A′={{1,2,3},{2,2,3},{3,2,3}},so $A^{\prime \prime }\subset A^{\prime },A^{\prime }\subset A^{\prime \prime },A^{\prime }=A^{\prime \prime }$
7. when one sample in the event $A^{\prime \prime \prime }$will definitely occur,$A^{\prime \prime \prime }$ can be called as certain event. if every sample in $A^{\prime \prime \prime }$ is impossible to occur,then $A^{\prime \prime \prime }$ can be called as impossible event.

references

1. 《数学》