Statistics and Probability with Applications for Engineers and Scientists Using MINITAB, R and JMP. Bhisham C. Gupta

Скачать книгу

alt="images"/> B's in n places.

      The number is usually called binomial coefficient, since it appears in the binomial expansion (for integer )

      (3.4.5)

      Example 3.4.5 The coefficient of in the expansion of is , since we can write as

      (3.4.6)

      The coefficient of is the number of ways to pick x of these factors and then choose a from each factor, while taking b from the remaining factors.

      3.4.4 Arrangements of n Objects Involving Several Kinds of Objects

Suppose that a collection of n objects are such that there are 's, 's, …, 's, where . Then total number of distinguishable arrangements of these several kinds of A's denoted by is

      (3.4.7)

      For if we think of each of the n places being originally filled with objects of type A, there are ways of choosing 's to be replaced by 's. In each of these ways, there are ways of choosing 's to be replaced by 's. Hence, the number of ways of choosing 's and replacing them with 's and choosing from the remaining A's and replacing them with 's is . Continuing this argument and using equation (3.4.4) shows that the number of ways of choosing 's and replacing them with 's, A's and replacing them with 's, and so on until the last A's replaced with 's, is

      To illustrate the application of combinations to probability problems involving finite sample spaces, we consider the following example.

      Example 3.4.6 (Combinations and probability) If 13 cards are dealt from a thoroughly shuffled deck of 52 ordinary playing cards, the probability of getting five spades is

      Solution: This result holds because the number of ways of getting five spades from the 13 spades in the deck is , and the number of ways of getting 8 nonspades from the 39 nonspades in the deck is , and hence, the number of ways five spades and eight nonspades occurs in a hand of 13 cards is the product . This is the number of elements in the sample space constituting the event of “getting five spades in dealing 13 cards from a shuffled deck.” Since the sample space consists of equally likely sample points, each sample point is assigned the same probability . Hence, the probability of getting five spades in dealing 13 cards is

      PRACTICE PROBLEMS FOR SECTION 3.4

      1 In a certain clinical trial, a medical team wants to study four different doses of a new medication for cervical cancer in five patients. In how many different ways can the team select one dose of the medication and one of the patients?

      2 A small motel with nine rooms has three twin beds in two rooms, two twin beds in three rooms, and one twin bed in rest of the four rooms. In how many different ways can the manager of the motel assign these rooms to a group of 16 guests who told the manager that they have no preference about whom they share the room with?

      3 In how many ways can a class of 30 students select a committee from the class that consists of a president, a vice president, a treasurer, and a secretary (a) if any student may serve either of these roles but no student may serve in multiple roles and (b) if any student may serve in multiple roles?

      4 A multiple‐choice board exam consists of 15 questions, each question having four possible answers. In how many different ways can a candidate select one answer to each question?

      5 A

Скачать книгу