[Probability & Statistics] 2. Sample Space, Events, Counting Sample Points

Probability & Statistics

Chapter 2. Probability

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Section 2.1 Sample Space

 

Definition 2.1

Sample Space : the set of all possible outcome of a statistical experiment

Figure 2.1 Tree diagram for Example 2.2

 

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Section 2.2 Events

Definition 2.2

Event : a subset of a sample space

Definition 2.3

Comlement : an event A with respect to S is the subset of all elements of S that are not in A

 

Definition 2.4

Intersection : the event contating all elements that are common to A and B

Definition 2.5

Two events a and b are mutually exclusive, or disjoint, if (A∩B) = ∅, that is, if A and B have no elements in common.

Definition 2.6

The union of the two evnets A and B, denoted by the sybol A ∩ B, is the event containing all the elements that belong to A or B or both.

Figure 2.3 Events represented by various regions

 

Figure 2.4 Events of the sample space S

 

Figure 2.5 Venn diagram for Exercises 2.19 and 2.20

 

 

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Section 2.3 Counting Sample Points

Rule 2.1

Multiplication rule: if an operation can be performed in n1 ways, and for each of these a second operation can be performed in n2 ways, then the two operations can be performed together in n1n2 ways

   n1 + n1 + · · · + n1(n2 times ) = n1n2.

 

Rule 2.2

Example 2.16 ★

Q. Sam is going to assemble a computer by himself. He has the choice of chips from two brands, a hard drive from four, memory from three, and an accessory bundle from five local stores. How many different ways can Sam order the parts?

A. 2*4*3*5=120

 

 

Definition 2.7

A permutation is an arrangement of all or part of a set of object

 

 

Definition 2.8

 

Theorem 2.1

The number of permutation of n objects is n!

 

Theorem 2.2 ★

The number of permutation of n distinct objects taken r at a time is 

 

Q. In a contest, there are 10 participants. There are three –gold, silver,
and bronze prizes for the participants. What is the total number of cases of prize reception?

A. 10P3 = 10!/(10-3)! = 10*9*8 = 720

 

 

Theorem 2.3

The number of permutation of n objects arranged in a circle is (n-1)!.

 

Theorem 2.4

The number of distinct permutations of n things of which n1 are of one kind, n2 of a second kind, ..., nk of a kth kind is

Theorem 2.5

 

Theorem 2.6

 

 

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