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Chapter 15: Q21P (page 744)

The following problem arises in quantum mechanics (see Chapter 13, Problem7.21). Find the number of ordered triples of nonnegative integers a, b, c whose suma+b+c is a given positive integer n. (For example, ifn=2, we could have(a,b,c)=(2,0,0)or(2,0,2)or (0,0,2)or (0,1,1)or or (1,0,1)or (1,1,0).) Hint: Show that this is the same as the number of distinguishable distributions of n identical balls in3boxes, and follow the method of the diagram in Example 5.

Short Answer

Expert verified

The required value is given below.

M=C(n+2,n)

Step by step solution

01

Given Information

Non negative integers whose sum is a+b+c.

02

Definition of uniform sample spaces

If a given experiment's sample space is known to be uniform, the probability of an event can be calculated using the event sizes and the sample space.

03

Find the values

Let the sum be n=a+b+c.

Bose-Einstein techniques in distribution is given below.

M=C(R−1+K,K)

Where K is number of balls or total sum of variable.

R is the number of variable or number of boxes.

M=C(3−1+n,n)=C(n+2,n)

Hence the required value is mentioned below.

M=C(n+2,n)

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Most popular questions from this chapter

(a) Set up a sample space for the 5 black and 10 white balls in a box discussed above assuming the first ball is not replaced. Suggestions: Number the balls, say 1 to 5 for black and 6 to 15 for white. Then the sample points form an array something like (2.4), but the point 3,3 for example is not allowed. (Why?

What other points are not allowed?) You might find it helpful to write the

numbers for black balls and the numbers for white balls in different colors.

(b) Let A be the event “first ball is white” and B be the event “second ball is

black.” Circle the region of your sample space containing points favorable to

A and mark this region A. Similarly, circle and mark region B. Count the

number of sample points in A and in B; these are and . The region

AB is the region inside both A and B; the number of points in this region is

. Use the numbers you have found to verify (3.2) and (3.1). Also find

and and verify (3.3) numerically.

(c) Use Figure 3.1 and the ideas of part (b) to prove (3.3) in general.

(a) Acandy vending machine is out of order. The probability that you get a candybar (with or without return of your money) is12, the probability that you getyour money back (with or without candy) is 12, and the probability that youget both the candy and your money back is 112. What is the probability that youget nothing at all? Suggestion: Sketch a geometric diagram similar to Figure 3.1, indicate regions representing the various possibilities and their probabilities; then set up a four-point sample space and the associated probabilities of the points.

(b) Suppose you try again to get a candy bar as in part (a). Set up the 16-point

sample space corresponding to the possible results of your two attempts tobuy a candy bar, and find the probability that you get two candy bars (andno money back); that you get no candy and lose your money both times; thatyou just get your money back both times.

Assuming a normal distribution, find the limitsμ±hfor a 90%confidence interval; for a 95%confidence interval; for a 99%confidence interval. What percent confidence interval isμ±1.3σ?Hints: See Section8, Example3, and Problems,8.7,8.22 and8.23.

Suppose13people want to schedule a regular meeting one evening a week. What is the probability that there is an evening when everyone is free if each person is already busy one evening a week?

Set up an appropriate sample space for each of Problems 1.1 to 1.10 and use itto solve the problem. Use either a uniform or non-uniform sample space or try both.

In a box there are 2 white, 3 black, and 4 red balls. If a ball is drawn at random,what is the probability that it is black? That it is not red?
See all solutions

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