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ssrs 2008 r2 barcode font Basic Probability in Software
CHAPTER 1 Basic Probability QR Decoder In None Using Barcode Control SDK for Software Control to generate, create, read, scan barcode image in Software applications. Quick Response Code Drawer In None Using Barcode drawer for Software Control to generate, create QR Code JIS X 0510 image in Software applications. Conditional Probability
Decode Quick Response Code In None Using Barcode reader for Software Control to read, scan read, scan image in Software applications. QR Code ISO/IEC18004 Creator In Visual C# Using Barcode maker for Visual Studio .NET Control to generate, create QR Code ISO/IEC18004 image in VS .NET applications. Let A and B be two events (Fig. 13) such that P(A) 0. Denote by P(B u A) the probability of B given that A has occurred. Since A is known to have occurred, it becomes the new sample space replacing the original S. From this we are led to the definition P(B u A) ; or P(A> B) P(A) (17) (18) Creating QR Code In .NET Using Barcode creation for ASP.NET Control to generate, create QR Code 2d barcode image in ASP.NET applications. Encoding QR Code ISO/IEC18004 In Visual Studio .NET Using Barcode creator for VS .NET Control to generate, create QR image in .NET framework applications. P(A > B) ; P(A) P(B u A) Printing QR Code In Visual Basic .NET Using Barcode maker for .NET framework Control to generate, create QR Code ISO/IEC18004 image in VS .NET applications. Paint EAN13 In None Using Barcode maker for Software Control to generate, create EAN 13 image in Software applications. Fig. 13 Painting ANSI/AIM Code 128 In None Using Barcode drawer for Software Control to generate, create Code 128C image in Software applications. UCC128 Maker In None Using Barcode creation for Software Control to generate, create UCC.EAN  128 image in Software applications. In words, (18) says that the probability that both A and B occur is equal to the probability that A occurs times the probability that B occurs given that A has occurred. We call P(B u A) the conditional probability of B given A, i.e., the probability that B will occur given that A has occurred. It is easy to show that conditional probability satisfies the axioms on page 5. Make Bar Code In None Using Barcode generation for Software Control to generate, create bar code image in Software applications. Make Data Matrix ECC200 In None Using Barcode encoder for Software Control to generate, create Data Matrix ECC200 image in Software applications. EXAMPLE 1.13 Find the probability that a single toss of a die will result in a number less than 4 if (a) no other information is given and (b) it is given that the toss resulted in an odd number. Print USD8 In None Using Barcode generation for Software Control to generate, create Code 11 image in Software applications. Barcode Reader In Java Using Barcode decoder for Java Control to read, scan read, scan image in Java applications. (a) Let B denote the event {less than 4}. Since B is the union of the events 1, 2, or 3 turning up, we see by Theorem 15 that P(B) P(1) P(2) P(3) 1 6 1 6 1 6 1 2 Paint Bar Code In VS .NET Using Barcode creation for ASP.NET Control to generate, create barcode image in ASP.NET applications. Painting Code 128B In None Using Barcode printer for Font Control to generate, create Code 128 Code Set A image in Font applications. assuming equal probabilities for the sample points. (b) Letting A be the event {odd number}, we see that P(A) P(B u A) Data Matrix ECC200 Recognizer In VS .NET Using Barcode recognizer for Visual Studio .NET Control to read, scan read, scan image in VS .NET applications. Code 128A Maker In VS .NET Using Barcode creator for Reporting Service Control to generate, create Code128 image in Reporting Service applications. 3 6 1 2. Matrix 2D Barcode Creation In C#.NET Using Barcode creation for .NET framework Control to generate, create Matrix 2D Barcode image in .NET applications. Printing ANSI/AIM Code 39 In ObjectiveC Using Barcode encoder for iPhone Control to generate, create Code39 image in iPhone applications. Also P(A > B) 1 3. Then
P(A > B) P(A) 1>3 1>2
Hence, the added knowledge that the toss results in an odd number raises the probability from 1 > 2 to 2 > 3. Theorems on Conditional Probability
Theorem 19 For any three events A1, A2, A3, we have P(A1 > A2 > A3) P(A1) P(A2 u A1) P(A3 u A1> A2) (19) In words, the probability that A1 and A2 and A3 all occur is equal to the probability that A1 occurs times the probability that A2 occurs given that A1 has occurred times the probability that A3 occurs given that both A1 and A2 have occurred. The result is easily generalized to n events. Theorem 110 If an event A must result in one of the mutually exclusive events A1, A2, c, An, then P(A) P(A1) P(A u A1) P(A2 ) P(A u A2) c P(An ) P(A u An ) (20) Independent Events
If P(B u A) P(B), i.e., the probability of B occurring is not affected by the occurrence or nonoccurrence of A, then we say that A and B are independent events. This is equivalent to P(A > B) P(A)P(B) (21) as seen from (18). Conversely, if (21) holds, then A and B are independent.
CHAPTER 1 Basic Probability
We say that three events A1, A2, A3 are independent if they are pairwise independent: P(Aj > Ak ) and P(Aj)P(Ak ) P(A1 > A2 > A3) j2k where j, k 1, 2, 3 (22) (23) P(A1)P(A2 )P(A3 ) Note that neither (22) nor (23) is by itself sufficient. Independence of more than three events is easily defined. Bayes Theorem or Rule
Suppose that A1, A2, c, An are mutually exclusive events whose union is the sample space S, i.e., one of the events must occur. Then if A is any event, we have the following important theorem: Theorem 111 (Bayes Rule): P(Ak u A) P(Ak) P(A u Ak) (24) a P(Aj) P(A u Aj) This enables us to find the probabilities of the various events A1, A2, c, An that can cause A to occur. For this reason Bayes theorem is often referred to as a theorem on the probability of causes. Combinatorial Analysis
In many cases the number of sample points in a sample space is not very large, and so direct enumeration or counting of sample points needed to obtain probabilities is not difficult. However, problems arise where direct counting becomes a practical impossibility. In such cases use is made of combinatorial analysis, which could also be called a sophisticated way of counting.

