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The multiplication rule is useful for determining the probability of an event that depends on other events. For example, suppose that in semiconductor manufacturing the probability is 0.10 that a chip that is subjected to high levels of contamination during manufacturing causes a product failure. The probability is 0.005 that a chip that is not subjected to high contamination levels during manufacturing causes a product failure. In a particular production run, 20% of the chips are subject to high levels of contamination. What is the probability that a product using one of these chips fails Clearly, the requested probability depends on whether or not the chip was exposed to high levels of contamination. We can solve this problem by the following reasoning. For any event B, we can write B as the union of the part of B in A and the part of B in A . That is, B 1A B2 1A B2

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Through block play, children are confronted with man,y mental challenges having to do with measurement, equality, balance, shape, spatial relationships and physical properties. - Creativity and Play [I121

This result is shown in the Venn diagram in Fig. 2-14. Because A and A are mutually exclusive, A B and A B are mutually exclusive. Therefore, from the probability of the union of mutually exclusive events in Equation 2-2 and the Multiplication Rule in Equation 2-6, the following total probability rule is obtained.

B = (B E1) (B E2) (B E3) (B E4)

Total Probability Rule (two events)

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Recall that a classic codebook cipher uses books filled with codes to encrypt and decrypt messages. Such codebooks have some inherent weaknesses. For example, any known plaintext immediately gives away part of the codebook. Also, information about the codebook can be obtained based on a statistical analysis of the ciphertext, in much the same way that information leaks through a simple substitution (although far more data is required to attack a respectable codebook than a simple substitution). Due to the threat of such attacks, new codebooks would have to be issued on a regular basis. Modern block ciphers can be viewed as roughly the electronic equivalent of classic codebooks. In block ciphers, a block of n plaintext bits is encrypted to a block of n ciphertext bits. Provided that the same key is used, the same plaintext block will always be encrypted to the same plaintext block, and vice versa. This is analogous to a classic codebook, except that the book, which would contain the binary n-tuples, is virtual in the sense that the lookups are accomplished using an algorithm that computes the required bits. That is, the entries of the codebook are computed as needed instead of being stored in an actual book. In a block cipher, the key determines the codebook. Consequently, if we change the key, we have, in effect, switched codebooks. If the key is k bits, then the block cipher algorithm can be viewed as 2k codebooks, indexed by the key. Therefore, by occasionally changing the key, we can avoid the classic

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For any events A and B, P1B2 P1B A2 P1B A 2 P1B A2P1A2 P1B A 2P1A 2 (2-7)

EXAMPLE 2-21

codebook attack mentioned above. A generic block cipher is illustrated in Figure 4.1, where P, is the ith block of plaintext and C, is the corresponding block of ciphertext,. The inner workings of the block cipher can be complex, but when encrypting, the net effect, is simply to lookup the ciphertext block that corresponds to the given plaintext block in the specified codebook, where the codebook is determined by the key.

Consider the contamination discussion at the start of this section. Let F denote the event that the product fails, and let H denote the event that the chip is exposed to high levels of contamination. The requested probability is P(F), and the information provided can be represented as P1F H2 P1H2 From Equation 2-7, P1F2 0.1010.202 0.00510.802 0.0235 0.10 0.20 and and P1F H 2 P1H 2 0.005 0.80

which can be interpreted as just the weighted average of the two probabilities of failure. The reasoning used to develop Equation 2-7 can be applied more generally. In the development of Equation 2-7, we only used the two mutually exclusive A and A . However, the fact that A A S , the entire sample space, was important. In general, a collection of sets E1, E2, p , Ek such that E1 E2 p Ek S is said to be exhaustive. A graphical display of partitioning an event B among a collection of mutually exclusive and exhaustive events is shown in Fig. 2-15 on page 43.

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Total Probability Rule (multiple events)

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