Название: Probability and Statistical Inference
Автор: Robert Bartoszynski
Издательство: John Wiley & Sons Limited
Жанр: Математика
isbn: 9781119243823
isbn:
We can generalize these considerations as follows:
Definition 3.2.1 An ordered sequence of
Theorem 3.2.5 The number of permutations of
Proof: The argument here repeatedly uses the “operation” principle: the first choice can be made in
If
Corollary 3.2.6 The set of
(3.2)
distinct ways.
The product (3.2) occurs often and has a special symbol:
to be read “
For a reason that will become apparent later, we adopt the convention
(3.4)
Example 3.4
The letters I, I, I, I, M, P, P, S, S, S, S are arranged at random. What is the probability that the arrangement will spell MISSISSIPPI?
Solution
We can solve this problem treating the choices of consecutive letters as “operations.” The first operation must give the letter M; hence, there is only one way of choosing it. The next letter (out of the remaining 10) must be an I, and it can be selected in four ways. Proceeding in this way, the sequence of consecutive 11 choices leading to the word MISSISSIPPI can be performed in
In this solution, the letters are regarded as distinguishable, as if we had four letters
which is the same as (3.5).
Example 3.5 Birthday Problem
The following problem has a long tradition and appears in many textbooks. If
Solution
Here we make the following assumption: (1) all years have 365 days (i.e., leap years are disregarded), (2) each day is equally likely to be a birthday of a person (i.e., births occur uniformly throughout the year), and (3) no twins attend the party. To compute