STAT1301 P&S I Tutorial 1 By Joseph Dong 21SEP2010,

Assignment Box Location 5/F Red side, Meng Wah Complex Box No. 6 Website: 2

Given a set of 5 differently colored points, in how many ways can you choose a unique subset of 2 points? 3

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.) 4

end of 12xxx Observation 1: The first two columns now contain all 2-element subsets, with lots of duplications. Observation 2: The duplications are of two kinds: 12xxx 12xxx vs. 21xxx 5

end of 12xxx 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. 6

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. 7

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

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. 9

Choose 5000 from different objects. Flip a coin times and observe exactly 5000 heads. Toss a die 60 times and observe each number 10 times. 10

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. 11

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 positionsreal and ghostare available for the 3 bars? 12

See Problem 4 in the Handout. 13

ordering DistinguishedIndistinguished replacement with with at least 1 **|*|*|** without at least 1 ||**|****| without 14 The Grouping Problem Permutation Multiplication Principle Combination

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 Morgans Laws Venns Diagrams De Morgans Laws Complementation (not) Intersection ( and) Union (or) 15

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a) The set of female year-1 or year-2 students b) The set of female local students c) The set of year-1 male non-local students d) The set of year-3 female local students e) The set of year-1 or year-2 non-local female students. 17

BASIC EXAMPLE SUBSTANTIALLY MORE TECHNICAL EXAMPLE See Problem 5 in the Handout. See Problem 6 in the Handout. 18

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. 19

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. 20

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Laplaces 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? 23

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. 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. 24

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 25

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