# ssrs 2008 r2 barcode font The Axioms of Probability in Software Generate QR-Code in Software The Axioms of Probability

The Axioms of Probability
Read Denso QR Bar Code In None
Using Barcode Control SDK for Software Control to generate, create, read, scan barcode image in Software applications.
Generating QR-Code In None
Using Barcode generation for Software Control to generate, create QR Code JIS X 0510 image in Software applications.
Suppose we have a sample space S. If S is discrete, all subsets correspond to events and conversely, but if S is nondiscrete, only special subsets (called measurable) correspond to events. To each event A in the class C of events, we associate a real number P(A). Then P is called a probability function, and P(A) the probability of the event A, if the following axioms are satisfied. Axiom 1 For every event A in the class C, P(A) Axiom 2 For the sure or certain event S in the class C, P(S) P(A1 < A2 < c ) P(A1 < A2) 1 c (2) Axiom 3 For any number of mutually exclusive events A1, A2, c, in the class C, P(A1) P(A2) (3) 0 (1)
QR-Code Recognizer In None
Using Barcode decoder for Software Control to read, scan read, scan image in Software applications.
QR Code JIS X 0510 Generation In C#.NET
Using Barcode creator for VS .NET Control to generate, create Denso QR Bar Code image in .NET framework applications.
In particular, for two mutually exclusive events A1, A2, P(A1) P(A2) (4)
Make Denso QR Bar Code In .NET
Using Barcode printer for ASP.NET Control to generate, create QR Code image in ASP.NET applications.
Denso QR Bar Code Encoder In VS .NET
Using Barcode encoder for Visual Studio .NET Control to generate, create QR Code image in VS .NET applications.
Some Important Theorems on Probability
Generating QR In VB.NET
Using Barcode encoder for .NET Control to generate, create QR Code JIS X 0510 image in .NET applications.
UPCA Maker In None
Using Barcode creation for Software Control to generate, create UPC-A Supplement 2 image in Software applications.
From the above axioms we can now prove various theorems on probability that are important in further work. Theorem 1-1 If A1 ( A2, then P(A1) Theorem 1-2 For every event A, 0 P(A) P(\) 1, (5) P(A2) and P(A2 A1) P(A2) P(A1).
Make Data Matrix 2d Barcode In None
Using Barcode drawer for Software Control to generate, create ECC200 image in Software applications.
Barcode Generation In None
Using Barcode drawer for Software Control to generate, create bar code image in Software applications.
i.e., a probability is between 0 and 1. Theorem 1-3 0 (6) i.e., the impossible event has probability zero.
EAN / UCC - 13 Creation In None
Using Barcode generator for Software Control to generate, create EAN 13 image in Software applications.
Drawing EAN128 In None
Using Barcode generator for Software Control to generate, create UCC.EAN - 128 image in Software applications.
CHAPTER 1 Basic Probability
Create ISBN In None
Using Barcode maker for Software Control to generate, create ISBN - 10 image in Software applications.
Paint UPCA In Java
Using Barcode generator for Eclipse BIRT Control to generate, create Universal Product Code version A image in BIRT reports applications.
Theorem 1-4 If Ar is the complement of A, then P(Ar) Theorem 1-5 If A 1 P(A) c (7) A1 < A2 < c < An, where A1, A2, . . . , An are mutually exclusive events, then P(A) In particular, if A P(A1) P(A2) c P(An) (8)
Encode Linear 1D Barcode In C#
Using Barcode encoder for .NET Control to generate, create 1D Barcode image in .NET framework applications.
Encoding ANSI/AIM Code 39 In Java
Using Barcode drawer for BIRT reports Control to generate, create Code 3/9 image in BIRT applications.
S, the sample space, then P(A1) P(A < B) P(A2) P(An) 1 (9)
Create EAN13 In Java
Using Barcode creation for Java Control to generate, create EAN 13 image in Java applications.
Barcode Generation In Objective-C
Using Barcode maker for iPhone Control to generate, create barcode image in iPhone applications.
Theorem 1-6 If A and B are any two events, then P(A) P(B) P(A > B) (10)
Generating Bar Code In Java
Using Barcode generation for Java Control to generate, create bar code image in Java applications.
GS1 DataBar Expanded Maker In .NET
Using Barcode creator for VS .NET Control to generate, create GS1 DataBar Limited image in Visual Studio .NET applications.
More generally, if A1, A2, A3 are any three events, then P(A1 < A2 < A3) P(A1) P(A2) P(A3) P(A2 > A3) P(A3 > A1) (11) P(A1 > A2)
P(A1 > A2 > A3) Generalizations to n events can also be made. Theorem 1-7 For any events A and B, P(A) P(A > B) P(A > Br)
(12)
Theorem 1-8 If an event A must result in the occurrence of one of the mutually exclusive events A1, A2, . . . , An, then P(A) P(A > A1) P(A > A2) c P(A > An) (13)
Assignment of Probabilities
If a sample space S consists of a finite number of outcomes a1, a2, c , an, then by Theorem 1-5, P(A1) P(A2) c P(An) 1 (14) where A1, A2, c , An are elementary events given by Ai {ai}. It follows that we can arbitrarily choose any nonnegative numbers for the probabilities of these simple events as long as (14) is satisfied. In particular, if we assume equal probabilities for all simple events, then 1 n , k 1, 2, c, n and if A is any event made up of h such simple events, we have P(Ak) P(A) (15)
h (16) n This is equivalent to the classical approach to probability given on page 5. We could of course use other procedures for assigning probabilities, such as the frequency approach of page 5. Assigning probabilities provides a mathematical model, the success of which must be tested by experiment in much the same manner that theories in physics or other sciences must be tested by experiment.
EXAMPLE 1.12
A single die is tossed once. Find the probability of a 2 or 5 turning up. {1, 2, 3, 4, 5, 6}. If we assign equal probabilities to the sample points, i.e., if we assume that
The sample space is S the die is fair, then
P(1)
P(2)
P(6)
The event that either 2 or 5 turns up is indicated by 2 < 5. Therefore, P(2 < 5) P(2) P(5) 1 6 1 6 1 3