Enumerative Combinatorics , Naïve Set Theory, and Sample Space

Enumerative Combinatorics, Naïve Set Theory, andSample Space STAT1301 P&S I Tutorial 1 By Joseph Dong 21SEP2010, MB103@HKU

Info • Assignment Box Location • 5/F Red side, MengWah Complex • Box No. 6 • email: jdong@hku.hk • Website: http://hku.hk/jdong/teaching/stat1301

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

A Reason for • Use 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 way…Look for patterns … (Someone has done this and the result is formulated in the combinatorial number.)

A Reason for , cont’d 12345 12354 12435 12453 12534 12543  end of 12xxx ..... 54321 • Observation 1: • The first two columns now contain all2-element subsets, with lots of duplications. • Observation 2: • The duplications are of two kinds: • 12xxx • 12xxx vs. 21xxx

A Reason for, cont’d 12345 12354 12435 12453 12534 12543  end of 12xxx ..... 54321 • Observation 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”.

A Reason for , cont’dAlternative Perspective of The Problem • Original: • 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.

Multinomial 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”? • In how many ways can you arrange 1 red ball, 1 yellow ball, 2 green balls, 3 blue balls, and 3 white balls in a line? in a circle? • The Multinomial Coefficient

Two Fundamental Principles of Counting Multiplication Principle Symmetry 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.

Thinking On the Executive Layer • Choose 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.

The Grouping Problem • Judy 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 becomes • In how many ways can you insert 3 bars in between the *s. • Observation: 6 slits to be occupied by 3 bars.

The Grouping Problem Generalized • What 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 positions—real and ghost—are available for the 3 bars?

A Grouping Problem in Disguise See Problem 4 in the Handout.

The Matrix of Counting Techniques The Grouping Problem Multiplication Principle Permutation Combination

Naïve Set Theory • Set 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 Morgan’s Laws • Venn’s Diagrams

Using The Set Language • Let denote the totality of students at the University of Hong Kong and , , , , the sets of year-1, year-2, year-3, and year-4 students respectively. Moreover, let denote the set of female students and the set of Non-local students. Express in words each of the following sets:

The set of female year-1 or year-2 students • The set of female local students • The set of year-1 male non-local students • The set of year-3 female local students • The set of year-1 or year-2 non-local female students.

Set Algebra Examples Basic Example Substantially more Technical Example • See Problem 5 in the Handout. • See Problem 6 in the Handout.

Sample Space and Event • A 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.

Vocabulary (Incomplete List) SET THEORETICAL LANGUAGE LOGICAL 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 .

Using the Language of Events • Express each of the following events in terms of the events , , and , and the operations of complementation, union, and intersection: • At least one of the events , , and occurs; • At most one of the events , , and occurs; • None of the events , , and occurs; • All three events occur; • Exactly one of the events , , and occurs; • and occur but not ; • occurs, if not then does not occur either

When is counting techniques used? • Laplace’s classical definition of Probability: • Involve counting the number of elements of both sets • Example • See Problem 2 in the handout. • Which outcomes are favorable? • What is the entire sample space?

A Probability is a Measure • Two 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. • Kolmogorov’s 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.

The Art of Identifying Sample Space and Events • Example • See Problem 37 of Assignment 1 • What is a good sample space to work on? • Example • See The Monty Hall Problem / Three Prisoners Problem • What two events are involved when the host opens box B which is known to him to be empty? • Event 1: the host chooses box B • Event 2: box B is empty

Importance of Fixing a Sample Space: Random Chord Paradox • A chord is randomly drawn from a unit circle. What’s the probability that the chord’s length is greater than ? • A Java Demo for 3 possible answers : , , and in that order: (usage: input a number) http://hku.hk/jdong/probability/RandomChord.jar (Src: http://hku.hk/jdong/probability/RandomChord.zip) • Why three answers?--They rely on different sample space specifications.

Enumerative Combinatorics , Naïve Set Theory, and Sample Space