STAT1301 P&S I Tutorial 1

Enumerative Combinatorics, Nave Set Theory, andSample SpaceSTAT1301 P&S ITutorial 1By Joseph Dong

21SEP2010, MB103@HKU 1InfoAssignment Box Location5/F Red side, Meng Wah ComplexBox No. 6 email: jdong@hku.hk Website: http://hku.hk/jdong/teaching/stat1301

22Appetizer Given a set of 5 differently colored points, in how many ways can you choose a unique subset of 2 points?

33Use the numbers set {1,2,3,4,5} to replace the colored points set. The Experiment is to choose a subset of 2 numbers, e.g., {1,2}.The Question is to find the number of different 2-element subsets can be chosen. The default method is enumeration: {1,2}, {1,3}, {1,4}, then count.This can go inefficient easily when large quantities are involved.Need a smart wayLook for patterns (Someone has done this and the result is formulated in the combinatorial number.)44123451235412435124531253412543 end of 12xxx.....54321Observation 1: The first two columns now contain all 2-element subsets, with lots of duplications.Observation 2:The duplications are of two kinds:12xxx12xxx vs. 21xxx55123451235412435124531253412543 end of 12xxx.....54321Observation 3: #duplications of the first 12xxx type depends on how many elements are there in the tail xxx.Observation 4:#duplications of the second type (12xxx vs 21xxx) depends on how many elements are there in the head 12.66Original:In how many ways can one choose from a set of 5 elements a subset of 2 elements.Alternative:In how many ways can one partition a set of 5 elements in to 2 groups, one of which containing 2 elements, the other 3.

77Multinomial Coefficients Can You Feel The Relationship?In how many ways can you partition the set {1,2,,10} in to 5 subsets consisting of 1,1, 2,3,and 3 elements respectively? In how many ways can you arrange the letters of the word STATISTICS?8

8Two Fundamental Principles of CountingMultiplication PrincipleSymmetry Argument (Indifference Principle)The Art of Identifying Symmetric Duplications

You need to be both good at thinking on this fundamental layer and thinking on the higher executive layer.99Thinking On the Executive LayerChoose 5000 from 20000 different objects.Flip a coin 20000 times and observe exactly 5000 heads.

Toss a die 60 times and observe each number 10 times.10

10The Grouping ProblemJudy has 7 identical chocolate beans and she wants to consume them in the next 4 days with the requirement at least 1 each day. In how many ways can she accomplish this?

* is a chocolate bean.* * * * * * ** *|* *|*|* *The problem becomesIn how many ways can you insert 3 bars in between the *s.Observation: 6 slits to be occupied by 3 bars.

1111The Grouping Problem GeneralizedWhat if Judy allow eating no bean for any day but still need to finish all 7 beans in 4 days?* * *||* * * *|There are effectively #(*) + #(|) positions for the 3 bars (|) to choose.We look at a simplified case: * * and |||Now think dynamically,Initial arrangement: | | | * *Now think of the dynamic process of morphing the initial arrangement into the following arrangement: | * | * |Then ask yourself how many positionsreal and ghostare available for the 3 bars?1212A Grouping Problem in DisguiseSee Problem 4 in the Handout.1313The Matrix of Counting TechniquesorderingDistinguishedIndistinguishedreplacementwithwith at least 1**|*|*|**

without at least 1||**|****|

without14

The Grouping ProblemPermutationMultiplication PrincipleCombination14Nave Set TheorySet Theory is the language of Mathematical Logic.The Twin Objects in Set Theory: Set vs. Elements(points) vs. The Triad of Set Operations:Complementation (Not)Union (Or)Intersection (And)

De Morgans Laws

Venns Diagrams1515Using The Set Language1616The set of female year-1 or year-2 studentsThe set of female local studentsThe set of year-1 male non-local studentsThe set of year-3 female local studentsThe set of year-1 or year-2 non-local female students.1717Set Algebra Examples Basic ExampleSubstantially more Technical ExampleSee Problem 5 in the Handout.See Problem 6 in the Handout.1818Sample Space and EventA sample space is a set.Results from Set Theory are applicable to Sample Space.A subset of a sample space is called an event.

The elements (points) of a sample space are called outcomes. The sample space is the set of all possible outcomes of a given random experiment.1919Vocabulary (Incomplete List)SET THEORETICAL LANGUAGELOGICAL MEANING IN TERMS OF EVENTS realizes A

A and B are incompatible

A implies B

A and B are both realized

One and only one of the events A and B is realized

One and only one of the events A1, A2, A3 is realized by any outcome/sample .20

20Using the Language of Events 21212222When is counting techniques used? Laplaces classical definition of Probability:

Involve counting the number of elements of both sets

ExampleSee Problem 2 in the handout.Which outcomes are favorable?What is the entire sample space?23

23A Probability is a MeasureTwo views of Probability:

Mathematical View: Probability as Count of elements, Length of a segment, Area of a surface, and Measure of a (measurable) set.

Physical View: Probability as Mass of a set of point masses, Mass of a line, of a surface, of a volume, etc.Kolmogorovs Axioms of Probability:Every event happens with a probability, and we only use numbers from [0,1] to quantify probability.The sure event happens with probability 100%.The sum of the probability of the happening of two (or a countable number of) disjoint events must be equal to the probability of any of them happening.2424The Art of Identifying Sample Space and EventsExampleSee Problem 37 of Assignment 1What is a good sample space to work on?

ExampleSee The Monty Hall Problem / Three Prisoners ProblemWhat two events are involved when the host opens box B which is known to him to be empty?Event 1: the host chooses box BEvent 2: box B is empty25

25Importance of Fixing a Sample Space: Random Chord Paradox2626