Probability and Statistical Inference. Robert Bartoszynski

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СКАЧАТЬ Rule) If and are finite sets consisting, respectively, of and elements, then the Cartesian product consists of elements.

Theorem 3.2.2 allows for an immediate generalization, we can define Cartesian products of more than two sets. Thus, if are some sets, then their Cartesian product is the set of all ‐tuples with . By easy induction, Theorem 3.2.2 can now be generalized as follows:

Theorem 3.2.3 If are finite, with consisting of elements , then contains elements.

      Example 3.2

      The total number of possible initials consisting of three letters (name, middle name, family name) is . Each three‐letter initial is an element of the set , where is the alphabet, so . The total number of possible two‐ or three‐letter initials is the number of the elements in the union , equal to

Example 3.3 License Plates

Most states now use the system where a license plate has six symbols. One type (call it A) of such licenses has a prefix of three letters followed by a three‐digit number (e.g., CQX 786). Other states use system (call it B) with a two‐letter prefix, followed by a four‐digit number (e.g., KN 7207). Still other states use system C, a digit and two letters, followed by a three digit number (e.g., 2CP 412). In addition states allow (for a special fee) “personalized” plates such as CATHY3 or MYCAR. Disregarding the personalized plates, which type of the license plate system can register most cars?

      Solution

      Let and stand for the alphabet and for the set of 10 digits: Then, a license plate from system A can be regarded as an element of , while a license plate in system B is an element of the set . The numbers of elements in these Cartesian products are and . The ratio is , so in the state using an A system 2.6 times more cars can be registered than in the state with a B system.

      Regarding system C, the answer depends whether or not 0 is allowed in the prefix. If the plate such as 0HY 314 is not allowed (e.g., because the digit 0 can be confused with the letter O), then the number of possible license plates is only , which is 10% less than the number of plates possible in states using system B. If 0 is allowed as the first character, then the numbers of plates of types and are the same.

In Examples 3.1 through 3.3 the set of ways of performing the second operations is the same regardless of which option was selected for the first operation. However, Theorem 3.2.1 remains true if the sets of ways of performing the second operation depend on the choice of the first operation. In particular, we can think of the first and second operation as two consecutive choices of an element from the same set, without returning the chosen elements. If the set, say , has elements, then the first operation (choice of an element) can be performed in ways. If the chosen element is not returned, then the second choice can be performed in ways only, and we have the following:

      Corollary 3.2.4 The number of ordered pairs with that can be formed out of distinct elements of a set of size is .

Instead of thinking in terms of operations, we can still use Cartesian products here. Thus, has СКАЧАТЬ