BASIC CONCEPTS IN PROBABILITY

1 BASIC CONCEPTS IN PROBABILITY . B. We see that the theory of PROBABILITY is at bottom only common sense reduced to calculation; it makes us appreciate with exactitude what reasonable minds feel by a sort of instinct, often without being able to account for it. It is remarkable that this science, which originated in the consideration of games of chance, should become the most important object of human knowledge. Laplace This chapter covers fundamental topics on probabilities of events. Main Topics q basics of Set Theory q Fundamental CONCEPTS in PROBABILITY q Conditional PROBABILITY q Independent Events q Total PROBABILITY Theorem and Bayes' Rule q Combined Experiments and Bernoulli Trials The materials covered in this chapter are essential for the study of the remaining chapters.

2 Emphasis should be put on the understanding of CONCEPTS and how they can be applied. 9. basics of Set Theory basics of Set Theory BASIC Definitions q set = a collection of objects, denoted by an upper case Latin letter Example: $ I J ' IG G G J. q element = an object in a set, denoted by a lower case Latin letter We say D is an element of $, D is in $, or D belongs to $, denoted as D $. q empty set = null set = a set with no elements , denoted by . q space = the set with all the elements for the problem under consideration (sometimes called universal set), denoted by 6.

3 Convention: Upper case Latin letter set Lower case Latin letter element If every element of set $ is also an element of set %, then $ is said to be a subset of %, denoted as $ | % or % } $. Set $ is said to be equal to set % if $ | % and $ } %, denoted as $ %. In this case, $ and % have exactly the same elements . Two sets are said to be disjoint if they do not have any element in common. Example : Consider the set of all positive integers smaller than 7: rule method $ I[ [ [ an integerJ. tabular method $ I J.]]]

4 The space (universal set) of a 6-face die Tabular form is not universally applicable. Example : Consider the set of all positive numbers smaller than 6: rule method % I[ [ [ a real numberJ. There is no tabular form for this set because it is uncountable. Example : Consider the set of all positive integers: & I[ [ ! [ an integerJ I J. Example : The set of human genders * = Ifemale, maleJ. 10. basics of Set Theory BASIC Set Operations Definitions: q The set of all elements of $ or % is called the union (or sum) of $ and %, denoted as $ > % or $ %.]]]]]]

5 Union of disjoint sets $ and % may be denoted as $ @ %. Convention: $ or % = either $ or % or both.. q The set of all elements common to $ and % is called the intersection (or product) of $ and %, denoted as $ ? % or $%. q The set of all elements of $ that are not in % is called the difference of $. and %, denoted as $ b %. q The set of all elements in the space 6 but not in $ is called the complement of $, denoted as $. It is equal to 6 b $. A simple and instructive way of illustrating the relationships among sets is the so-called Venn diagram, as illustrated below.

6 $ . $ .. $ > % $ ? % .. % . % .. 6 6.. $ .. $.. $ .. $ b % . % .. 6 . 6.. Figure : BASIC set operations. 12. basics of Set Theory Example : Set Operations For $, %, and & considered in Examples , , and : $|&. $>% I[ [ [ a real numberJ. $>& &. %>& I[ [ a positive integer or a real number satisfying [ J. This set has a mixed type. $?% I J. $?& $. %?& I J. $b% I J. $b& . %b$ I[ [ [ a noninteger real numberJ. %b& I[ [ [ a noninteger real numberJ. & b$ I[ [ w [ an integerJ I J. & b% I[ [ w [ an integerJ I J.]]]]]]]]]]]]]]]]]]

7 Space 6 depends on what we are considering. If we are considering only positive real numbers, then 6 I[ [ ! [ realJ. Thus, $ I[ [ a positive real number other than J. % I[ [ w [ a real numberJ. & I[ [ a noninteger positive real numberJ. If, however, we are considering all real numbers, then 6 I[ [ realJ. Thus $ I[ [ a real number other than J. % I[ [ or [ w [ a real numberJ. & I[ [ or [ a noninteger positive real numberJ. 13. basics of Set Theory BASIC Algebra of Sets Algebra of sets Algebra of numbers Union > sum.]]]]]]]]]]]]]]]]]]]]]

8 Intersection ? product c . 1 $>% %>$ D E E D. 2 $?% %?$ DcE EcD. 3 $> %>& $>%>& D E F D E F. 4 $? %?& $?%?& Dc EcF DcEcF. 5 $? %>& $?% > $?& Dc E F DcE DcF. 6 $> %?& $>% ? $>& see below Since $ ? $ $, $ ? % | $, $ ? & | $, and D E D F DcD DcE DcF EcF. Line 6 in the table above follows from $>% ? $>& $?$ > $?% > $?& > %?& $> %?&. _ ^] `. $. This illustrates that set algebra has its own rules. De Morgan's laws: $>% $?% ( ). $?% $>% ( ). Similarly, $>%>& $>' $?' $?%>& $? %?& $?%?&. $ > c c c > $Q $ ? c c c ?

9 $Q. $ ? c c c ? $Q $ > c c c > $Q. $ > $ ? $ > $ $ ? $ > $ ? $ . Rules: (1) interchange > and ?; (2) interchange e and e . However, care should be taken when dealing with multiple nests, as demonstrated below. Example : $. _ ? ^] %` > & '>& '?& $?% ?& $>%?& ( ). '. 14. Fundamental CONCEPTS in PROBABILITY Fundamental CONCEPTS in PROBABILITY Definitions q random experiment = experiment (action) whose result is uncertain (cannot be predicted with certainty) before it is performed q trial = single performance of the random experiment q outcome = result of a trial q sample space 6 = the set of all possible outcomes of a random experiment q event = subset of the sample space 6 (to which a PROBABILITY can be assigned).

10 = a collection of possible outcomes q sure event = sample space 6 (an event for sure to occur). q impossible event = empty set (an event impossible to occur). We say an event has occurred if and only if the outcome observed belongs to the set of the event, as explained below. Example : Die-Rolling Events Rolling a die is a random experiment. An outcome can be any number from 1. to 6. Sample space = I J. Some possible events are $ Ian even number shows upJ = I J (3 outcomes). % Ia number greater than 5 shows upJ.